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Metamath Proof Explorer

Theorem raleqbii

Description: Equality deduction for restricted universal quantifier, changing both formula and quantifier domain. Inference form. (Contributed by David Moews, 1-May-2017)

	Ref	Expression
Hypotheses	raleqbii.1	$⊢ A = B$
	raleqbii.2	$⊢ ψ \leftrightarrow χ$
Assertion	raleqbii	$⊢ \forall x \in A ψ \leftrightarrow \forall x \in B χ$

Proof

Step	Hyp	Ref	Expression
1		raleqbii.1	$⊢ A = B$
2		raleqbii.2	$⊢ ψ \leftrightarrow χ$
3	1	eleq2i	$⊢ x \in A \leftrightarrow x \in B$
4	3 2	imbi12i	$⊢ (x \in A \to ψ) \leftrightarrow (x \in B \to χ)$
5	4	ralbii2	$⊢ \forall x \in A ψ \leftrightarrow \forall x \in B χ$