Theorem cbvmovw

Description: Change bound variable. Uses only Tarski's FOL axiom schemes. See cbvmo and cbvmow for versions with fewer disjoint variable conditions but requiring more axioms. (Contributed by NM, 9-Mar-1995) (Revised by GG, 30-Sep-2024)

	Ref	Expression
Hypothesis	cbvmovw.1	$$⊢x=y\to \left(\phi \leftrightarrow \psi \right)$$
Assertion	cbvmovw	$$⊢{\exists }^{*}x\phi \leftrightarrow {\exists }^{*}y\psi$$

Proof

Step	Hyp	Ref	Expression
1		cbvmovw.1	$$⊢x=y\to \left(\phi \leftrightarrow \psi \right)$$
2		equequ1	$$⊢x=y\to \left(x=z\leftrightarrow y=z\right)$$
3	1 2	imbi12d	$$⊢x=y\to \left(\left(\phi \to x=z\right)\leftrightarrow \left(\psi \to y=z\right)\right)$$
4	3	cbvalvw	$$⊢\forall x\left(\phi \to x=z\right)\leftrightarrow \forall y\left(\psi \to y=z\right)$$
5	4	exbii	$$⊢\exists z\forall x\left(\phi \to x=z\right)\leftrightarrow \exists z\forall y\left(\psi \to y=z\right)$$
6		dfmo	$$⊢{\exists }^{*}x\phi \leftrightarrow \exists z\forall x\left(\phi \to x=z\right)$$
7		dfmo	$$⊢{\exists }^{*}y\psi \leftrightarrow \exists z\forall y\left(\psi \to y=z\right)$$
8	5 6 7	3bitr4i	$$⊢{\exists }^{*}x\phi \leftrightarrow {\exists }^{*}y\psi$$