I am looking for a proof of
$x=y \implies x+z=y+z$
but i couldn't deduce it from equivalence and addition axioms. I've looked through on internet but the only thing i've found was something like that:
$x=x \implies x+z=x+z \implies x+z = y+z$ which means you can substitute $x$ and $y$. But it seems saying that you can substitute $x$ and $y$ is the same thing with saying that $x=y \implies x+z=y+z$. And this is what we want to prove and what we presume before proving it, so it proves itself.
Is
$(x=x \implies x+z=x+z \implies x+z = y+z)$ proof of $x=y \implies x+z=y+z$ and if it is not, what is the proof? How could one deduce it from addition and equivalence axioms?
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1You have to use the logical axioms for equality. – Mauro ALLEGRANZA May 09 '18 at 11:13
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1See the very similar post : how do we prove that if $f(x)=g(x)$ then $f(x)+h(x)=g(x)+h(x)$ ?. – Mauro ALLEGRANZA May 09 '18 at 11:18