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Metamath Proof Explorer

Theorem ralbidva

Description: Formula-building rule for restricted universal quantifier (deduction form). (Contributed by NM, 4-Mar-1997) Reduce dependencies on axioms. (Revised by Wolf Lammen, 29-Dec-2019)

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

Proof

Step	Hyp	Ref	Expression
1		ralbidva.1	$⊢ (φ \land x \in A) \to (ψ \leftrightarrow χ)$
2	1	pm5.74da	$⊢ φ \to ((x \in A \to ψ) \leftrightarrow (x \in A \to χ))$
3	2	ralbidv2	$⊢ φ \to (\forall x \in A ψ \leftrightarrow \forall x \in A χ)$