Theorem bj-chvarv 33260

Description: Version of chvar 2417 with a disjoint variable condition, which does not require ax-13 2391. (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 221	. . 3 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))
4	1, 3	spimv1 2298	. 2 ⊢ (∀𝑥𝜑 → 𝜓)
5		bj-chvarv.2	. 2 ⊢ 𝜑
6	4, 5	mpg 1898	1 ⊢ 𝜓

Syntax hints:   → wi 4   ↔ wb 198  Ⅎwnf 1884

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-12 2222

This theorem depends on definitions:  df-bi 199  df-ex 1881  df-nf 1885

This theorem is referenced by:  bj-axrep2  33315  bj-axrep3  33316