Theorem bj-chvarv 32850

Description: Version of chvar 2298 with a dv condition, which does not require ax-13 2282. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
bj-chvarv.nf	⊢ Ⅎ𝑥𝜓
bj-chvarv.1	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
bj-chvarv.2	⊢ 𝜑

Assertion

Ref	Expression
bj-chvarv	⊢ 𝜓

Distinct variable group:   𝑥,𝑦

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

Proof of Theorem bj-chvarv

Step	Hyp	Ref	Expression
1		bj-chvarv.nf	. . 3 ⊢ Ⅎ𝑥𝜓
2		bj-chvarv.1	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
3	2	biimpd 219	. . 3 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))
4	1, 3	spimv1 2153	. 2 ⊢ (∀𝑥𝜑 → 𝜓)
5		bj-chvarv.2	. 2 ⊢ 𝜑
6	4, 5	mpg 1764	1 ⊢ 𝜓

Syntax hints:   → wi 4   ↔ wb 196  Ⅎwnf 1748

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-12 2087

This theorem depends on definitions:  df-bi 197  df-ex 1745  df-nf 1750

This theorem is referenced by:  bj-axrep2  32914  bj-axrep3  32915