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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 → ( 𝜑 ↔ 𝜓 ) )
	chvar.3	⊢ 𝜑
Assertion	chvar	⊢ 𝜓

Proof

Step	Hyp	Ref	Expression
1		chvar.1	⊢ Ⅎ 𝑥 𝜓
2		chvar.2	⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
3		chvar.3	⊢ 𝜑
4	2	biimpd	⊢ ( 𝑥 = 𝑦 → ( 𝜑 → 𝜓 ) )
5	1 4	spim	⊢ ( ∀ 𝑥 𝜑 → 𝜓 )
6	5 3	mpg	⊢ 𝜓