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Metamath Proof Explorer

Theorem raleqbidvv

Description: Version of raleqbidv with additional disjoint variable conditions, not requiring ax-8 nor df-clel . (Contributed by BJ, 22-Sep-2024)

	Ref	Expression
Hypotheses	raleqbidvv.1	$⊢ φ \to A = B$
	raleqbidvv.2	$⊢ φ \to (ψ \leftrightarrow χ)$
Assertion	raleqbidvv	$⊢ φ \to (\forall x \in A ψ \leftrightarrow \forall x \in B χ)$

Step	Hyp	Ref	Expression
1		raleqbidvv.1	$⊢ φ \to A = B$
2		raleqbidvv.2	$⊢ φ \to (ψ \leftrightarrow χ)$
3	2	adantr	$⊢ (φ \land x \in A) \to (ψ \leftrightarrow χ)$
4	1 3	raleqbidva	$⊢ φ \to (\forall x \in A ψ \leftrightarrow \forall x \in B χ)$