Theorem bj-chvarvv 31913

Description: Version of chvarv 2251 with a dv condition, which does not require ax-13 2234. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)

Hypotheses

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

Assertion

Ref	Expression
bj-chvarvv	⊢ 𝜓

Distinct variable groups:   𝑥,𝑦   𝜓,𝑥

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

Proof of Theorem bj-chvarvv

Step	Hyp	Ref	Expression
1		bj-chvarvv.1	. . 3 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
2	1	bj-spvv 31910	. 2 ⊢ (∀𝑥𝜑 → 𝜓)
3		bj-chvarvv.2	. 2 ⊢ 𝜑
4	2, 3	mpg 1715	1 ⊢ 𝜓

Syntax hints:   → wi 4   ↔ wb 195

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875

This theorem depends on definitions:  df-bi 196  df-ex 1696

This theorem is referenced by:  bj-axext3  31957  bj-axrep1  31976  bj-axsep2  32113