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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 → ( 𝜑 ↔ 𝜓 ) )
	chvarfv.2	⊢ 𝜑
Assertion	chvarfv	⊢ 𝜓

Proof

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