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Metamath Proof Explorer

Theorem chvar

Description: Implicit substitution of y for x into a theorem. Usage of this theorem is discouraged because it depends on ax-13 . Use the weaker chvarfv if possible. (Contributed by Raph Levien, 9-Jul-2003) (Revised by Mario Carneiro, 3-Oct-2016) (New usage is discouraged.)

	Ref	Expression
Hypotheses	chvar.1	$⊢ Ⅎ x ψ$
	chvar.2	$⊢ x = y \to (φ \leftrightarrow ψ)$
	chvar.3	$⊢ φ$
Assertion	chvar	$⊢ ψ$

Proof

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