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Metamath Proof Explorer

Theorem rexbidva

Description: Formula-building rule for restricted existential quantifier (deduction form). (Contributed by NM, 9-Mar-1997) Reduce dependencies on axioms. (Revised by Wolf Lammen, 6-Dec-2019) (Proof shortened by Wolf Lammen, 10-Dec-2019)

		Ref	Expression
	Hypothesis	ralbidva.1	$⊢ (φ \land x \in A) \to (ψ \leftrightarrow χ)$
	Assertion	rexbidva	$⊢ φ \to (\exists x \in A ψ \leftrightarrow \exists x \in A χ)$

Proof

Step	Hyp	Ref	Expression
1		ralbidva.1	$⊢ (φ \land x \in A) \to (ψ \leftrightarrow χ)$
2	1	pm5.32da	$⊢ φ \to ((x \in A \land ψ) \leftrightarrow (x \in A \land χ))$
3	2	rexbidv2	$⊢ φ \to (\exists x \in A ψ \leftrightarrow \exists x \in A χ)$