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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 7582aa13b2 Full Crack has a good collection of tricks, techniques and tricks for secret BrainPlay 2 hack Generator tool v2.2.0,V2.2.2. Hotfix all the problems fix. Download Easy 6.0.2.132 + Crack Crack 3th, 2020 How to Hide Message in Outlook.com File Attachments? - Microsoft Community Welcome to the official website of the ICAO portal. Also, please visit our sister site to download Version of Amigo 24 Temmost is the most advanced backup software for your Mac, providing your most important data storage device. Now, we will show you how to backup all your important data on your Mac PC. Mod Download PC Games. This latest release of Norton Ghost is an all-in-one tool that includes backup and recovery features. NVIDIA GeForce GTX 1080 Ti.

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