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Metamath Proof Explorer

Theorem rspcv

Description: Restricted specialization, using implicit substitution. (Contributed by NM, 26-May-1998) Drop ax-10 , ax-11 , ax-12 . (Revised by SN, 12-Dec-2023)

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

Proof

Step	Hyp	Ref	Expression
1		rspcv.1	$⊢ x = A \to (φ \leftrightarrow ψ)$
2		id	$⊢ A \in B \to A \in B$
3	1	adantl	$⊢ (A \in B \land x = A) \to (φ \leftrightarrow ψ)$
4	2 3	rspcdv	$⊢ A \in B \to (\forall x \in B φ \to ψ)$