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	$⊢ x = y \to (φ \leftrightarrow ψ)$
	Assertion	cbvrexvw	$⊢ \exists x \in A φ \leftrightarrow \exists y \in A ψ$

Proof

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

Metamath Proof Explorer

Theorem cbvrexvw

Proof