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Metamath Proof Explorer

Theorem spvv

Description: Specialization, using implicit substitution. Version of spv with a disjoint variable condition, which does not require ax-7 , ax-12 , ax-13 . (Contributed by NM, 30-Aug-1993) (Revised by BJ, 31-May-2019)

		Ref	Expression
	Hypothesis	spvv.1	$⊢ x = y \to (φ \leftrightarrow ψ)$
	Assertion	spvv	$⊢ \forall x φ \to ψ$

Proof

Step	Hyp	Ref	Expression
1		spvv.1	$⊢ x = y \to (φ \leftrightarrow ψ)$
2	1	biimpd	$⊢ x = y \to (φ \to ψ)$
3	2	spimvw	$⊢ \forall x φ \to ψ$