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Do all equivalent sentences have same truth value?
In my book the following statement says:
Let:
$$\phi=\varphi_1 \wedge \varphi_2 \ \wedge \ \cdots \ \wedge \ \varphi_n$$
$$\psi=\varphi_1 \vee \varphi_2 \ \vee \ \cdots \ \vee \ \varphi_n$$
For all sentences $\varphi$, $T_1 \vdash \phi \leftrightarrow \psi$.
$T_1$: $\text{provable in PA}$
On the other hand, this statement says:
$$\phi=\varphi_1 \wedge \varphi_2 \ \wedge \ \cdots \ \wedge \ \varphi_n$$
$$\psi=\varphi_1 \vee \varphi_2 \ \vee \ \cdots \ \vee \ \varphi_n$$
$\text{Provable in ZF}$
For all sentences $\varphi$, $T_1 \vdash \phi \leftrightarrow \psi$.
$T_1$: $\text{provable in ZF}$
Can you explain why the first proposition is not provable in PA but the second one is provable in ZF?
A:
You have two separate theorems which, when combined, say that for every sentence $\varphi$, if $\varphi$ is provable in one of two theories, then $\varphi$ is provable in the other theory. The first theorem says that if $\varphi$ is provable in PA, then $\psi$ is also provable in PA. The second theorem says that if $\varphi$ is provable in ZF, then
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