This is a statement that is logically incorrect.
Let’s suppose you have two statements: “listen and follow” and “receive eternal life”. In logical terms, let’s call “listen and follow”
P and “receive eternal life”
Q.
Now, the logical statement ‘if P then Q’ (or, symbolically,
P → Q) is the one you’re talking about. You’re claiming that
P → Q has the same value as ‘if not-P then not-Q’ (or, symbolically,
~P → ~Q). That is patently false. I could give you the truth table, but an example will suffice.
Let’s suppose that P is “Fluffy is a dog” and Q is “Fluffy is a mammal.” P->Q is then “if Fluffy is a dog, then Fluffy is a mammal.” That’s clearly true. However, you’re asserting that “if Fluffy is not a dog, then Fluffy is not a mammal.” That’s false. (Fluffy could be a cat.)
So, your logic fails here.
Moreover, you’ve created a “conditional” that isn’t in the text. You even quoted it for us:
The statement here is “you do not believe (me) because you do not belong to my sheep.” There’s another statement as well: “I give [my sheep] eternal life.” These two only
imply “if you are my sheep, then I will give you eternal life”, but they
do not imply “I will not give eternal life to those who are not my sheep.” (After all, the Catholic Church teaches that it is possible for those who through no fault of their own have not come to belief.)
The context of the thread is John 10:27-28. I just included context.
when I quoted (emphasis mine)
“
Jn 10:24 So the Jews gathered round him and said to him, “How long will you keep us in suspense? If you are the Christ, tell us plainly.” 25 Jesus answered them, “I told you, and you do not believe. The works that I do in my Father’s name, they bear witness to me; 26 but you do not believe, because you do not belong to my sheep. 27 My sheep hear my voice, and I know them, and they follow me; 28 and I give them eternal life, and they shall never perish, and no one shall snatch them out of my hand.”
The Jews Jesus is talking to, according to (Mt 15 below) are members of (“the lost sheep of the house of Israel”) .
These Jews wanted Jesus to specifically say He is the Christ, because they thought He was keeping them in suspense by not saying specifically He’s the “Christ”.
These Jews therefore, don’t believe Jesus is the Christ. So in extension, they don’t believe Jesus is their shepherd, and in reverse, Jesus is saying they aren’t His sheep.
So my analogies of “sheep” and shepherd, and each recognizing the other, and sheep recognizing the shepherd’s voice, and following the shepherd, and resulting conditions if that relationship happens or doesn’t happen remains viable.
consider an additional example,
**Matt 15: 22 **And behold, a Canaanite woman from that region came out and cried, “Have mercy on me, O Lord, Son of David; my daughter is severely possessed by a demon.” 23 But he did not answer her a word. And his disciples came and begged him, saying, “Send her away, for she is crying after us.” 24 He answered, “I was sent only to the lost sheep of the house of Israel.”[e] 25 But she came and knelt before him, saying, “Lord, help me.” 26 And he answered, “It is not fair to take the children’s bread and throw it to the dogs.” 27 She said, “Yes, Lord, yet even the dogs eat the crumbs that fall from their master’s table.” 28 Then Jesus answered her, “O woman, great is your faith! Be it done for you as you desire.” And her daughter was healed instantly.
Even after He said to her "“I was sent only to the lost sheep of the house of Israel” and “It is not fair to take the children’s bread and throw it to the dogs.”
She wasn’t phased. She hung in there under some formidable pressure.
So I ask, have any conditions previously mentioned in my posts, been overturned here?
She recognized Jesus for who He is, otoh, the Jews (lost sheep) in the story didn’t
She demonstrated faith in Him, the Jews ( lost sheep) in the story didn’t
She wouldn’t be shewed away from Jesus.

I’m not sure that could be said about the (lost sheep) Jews in the story
Jesus took care of her (this little lamb) and her utmost need because of her faith
If she was scared off would the good results she received, have happened? Those are all conditions, and they separate and distinguish sheep and shepherd
No question, Jesus was rough on her as were the apostles. Yet she wouldn’t leave Jesus.
Unlike the Jews (lost sheep) above who were in a fog, she was clear on who Jesus is.
She knelt before Him, she called Him Lord, master, and Son of David. The Jews didn’t do that.
This lamb who recognized her shepherd and would not go away from Him even when other forces tried to push her away she remained firm. Unlike the (lost sheep)
Go back and look at my previous posts and examples. They’re consistent, logical, and the necessary conditions for desired results were met.