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Metamath Proof Explorer

Theorem chvarfv

Description: Implicit substitution of y for x into a theorem. Version of chvar with a disjoint variable condition, which does not require ax-13 . (Contributed by Raph Levien, 9-Jul-2003) (Revised by BJ, 31-May-2019)

	Ref	Expression
Hypotheses	chvarfv.nf	$⊢ Ⅎ x ψ$
	chvarfv.1	$⊢ x = y \to (φ \leftrightarrow ψ)$
	chvarfv.2	$⊢ φ$
Assertion	chvarfv	$⊢ ψ$

Proof

Step	Hyp	Ref	Expression
1		chvarfv.nf	$⊢ Ⅎ x ψ$
2		chvarfv.1	$⊢ x = y \to (φ \leftrightarrow ψ)$
3		chvarfv.2	$⊢ φ$
4	2	biimpd	$⊢ x = y \to (φ \to ψ)$
5	1 4	spimfv	$⊢ \forall x φ \to ψ$
6	5 3	mpg	$⊢ ψ$