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In fact, I have two questions about universal specification, because I'm not sure about its application, so the next applications are okey: $$\forall x \forall y [R(x, z) \rightarrow \exists y R(f(w, y), y)] \vdash \forall z [R(f(w, y), z) \rightarrow \exists y R (f(w, y), y)] $$ and

$$\forall x \exists y R (x, y)\vdash \exists x R(x, x) $$ I think that the second one is incorrect, but I don't have any counterexample. Thanks for your answers.

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    $\begingroup$ The second one is incorrect, because the "Universal Specification" axioms says: "$∀x α → α[t/x]$, where $t$ is substitutable for $x$ in $α$" and $x$ is not substitutable for $y$ in $∃yR(x,y)$. $\endgroup$ Oct 14, 2020 at 7:15
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    $\begingroup$ Also the first one is wrong; in the premise $∀x∀y[R(x,z) → \ldots]$ we have that $z$ in $R(x,z)$ is free, while in the conclusion $∀z[R(f(w,y),z) → \ldots]$ it is bound. $\endgroup$ Oct 14, 2020 at 7:18
  • $\begingroup$ Thanks for your answers $\endgroup$ Oct 15, 2020 at 0:15
  • $\begingroup$ You are welcome :-) $\endgroup$ Oct 15, 2020 at 5:47

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