Show that the formula $\displaystyle x=y \rightarrow Pzfx \rightarrow Pzfy$ (where $\displaystyle f$ is a one-place function symbol and $\displaystyle P$ is a two-place predicate symbol) is valid.
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(*) If $\displaystyle \gamma;\alpha \models \phi$, then $\displaystyle \gamma \models (\alpha \rightarrow \phi)$.
We show $\displaystyle \models x=y \rightarrow Pzfx \rightarrow Pzfy$.
By (*), it suffices to show that $\displaystyle \{x=y, Pzfx\} \models Pzfy$. Therefore, we need to show every $\displaystyle A$ that satisfies $\displaystyle x=y$ and $\displaystyle Pzfx$ with every function $\displaystyle s:V \rightarrow |A|$ satisfies $\displaystyle Pzfy$ with $\displaystyle s$. That is, if $\displaystyle \models_Ax=y[s]$ and $\displaystyle \models_A Pzfx[s]$, then $\displaystyle \models_A Pzfy[s]$.
I am stuck here. Any help will be appreciated.