Theorem chvarvv 1908

Description: Version of chvarv 1937 with a disjoint variable condition. (Contributed by BJ, 31-May-2019.)

Hypotheses

Ref	Expression
chvarvv.1	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
chvarvv.2	⊢ 𝜑

Assertion

Ref	Expression
chvarvv	⊢ 𝜓

Distinct variable groups:   𝑥,𝑦   𝜓,𝑥

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑦)

Proof of Theorem chvarvv

Step	Hyp	Ref	Expression
1		chvarvv.1	. . 3 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
2	1	spvv 1907	. 2 ⊢ (∀𝑥𝜑 → 𝜓)
3		chvarvv.2	. 2 ⊢ 𝜑
4	2, 3	mpg 1451	1 ⊢ 𝜓

Syntax hints:   → wi 4   ↔ wb 105

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534

This theorem depends on definitions:  df-bi 117  df-nf 1461

This theorem is referenced by:  prodfdivap  11557