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Metamath Proof Explorer

Theorem ralbiia

Description: Inference adding restricted universal quantifier to both sides of an equivalence. (Contributed by NM, 26-Nov-2000)

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

Step	Hyp	Ref	Expression
1		ralbiia.1	$⊢ x \in A \to (φ \leftrightarrow ψ)$
2	1	pm5.74i	$⊢ (x \in A \to φ) \leftrightarrow (x \in A \to ψ)$
3	2	ralbii2	$⊢ \forall x \in A φ \leftrightarrow \forall x \in A ψ$