Theorem cbvrexvw

Description: Change the bound variable of a restricted existential quantifier using implicit substitution. Version of cbvrexv with a disjoint variable condition, which does not require ax-10 , ax-11 , ax-12 , ax-13 . (Contributed by NM, 2-Jun-1998) Avoid ax-13 . (Revised by GG, 10-Jan-2024)

	Ref	Expression
Hypothesis	cbvralvw.1	$$⊢x=y\to \left(\phi \leftrightarrow \psi \right)$$
Assertion	cbvrexvw	$$⊢\exists x\in A\phi \leftrightarrow \exists y\in A\psi$$

Proof

Step	Hyp	Ref	Expression
1		cbvralvw.1	$$⊢x=y\to \left(\phi \leftrightarrow \psi \right)$$
2		eleq1w	$$⊢x=y\to \left(x\in A\leftrightarrow y\in A\right)$$
3	2 1	anbi12d	$$⊢x=y\to \left(\left(x\in A\wedge \phi \right)\leftrightarrow \left(y\in A\wedge \psi \right)\right)$$
4	3	cbvexvw	$$⊢\exists x\left(x\in A\wedge \phi \right)\leftrightarrow \exists y\left(y\in A\wedge \psi \right)$$
5		df-rex	$$⊢\exists x\in A\phi \leftrightarrow \exists x\left(x\in A\wedge \phi \right)$$
6		df-rex	$$⊢\exists y\in A\psi \leftrightarrow \exists y\left(y\in A\wedge \psi \right)$$
7	4 5 6	3bitr4i	$$⊢\exists x\in A\phi \leftrightarrow \exists y\in A\psi$$