This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem rexeqbii

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

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

Proof

Step	Hyp	Ref	Expression
1		rexeqbii.1	$⊢ A = B$
2		rexeqbii.2	$⊢ ψ \leftrightarrow χ$
3	1	eleq2i	$⊢ x \in A \leftrightarrow x \in B$
4	3 2	anbi12i	$⊢ (x \in A \land ψ) \leftrightarrow (x \in B \land χ)$
5	4	rexbii2	$⊢ \exists x \in A ψ \leftrightarrow \exists x \in B χ$