Theorem bj-ceqsal 34980

Description: Remove from ceqsal 3457 dependency on ax-ext 2710 (and on df-cleq 2731, df-v 3425, df-clab 2717, df-sb 2073). (Contributed by BJ, 12-Oct-2019.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
bj-ceqsal.1	⊢ Ⅎ𝑥𝜓
bj-ceqsal.2	⊢ 𝐴 ∈ V
bj-ceqsal.3	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
bj-ceqsal	⊢ (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜓)

Distinct variable group:   𝑥,𝐴

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

Proof of Theorem bj-ceqsal

Step	Hyp	Ref	Expression
1		bj-ceqsal.2	. 2 ⊢ 𝐴 ∈ V
2		bj-ceqsal.1	. . 3 ⊢ Ⅎ𝑥𝜓
3		bj-ceqsal.3	. . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
4	2, 3	bj-ceqsalgv 34978	. 2 ⊢ (𝐴 ∈ V → (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜓))
5	1, 4	ax-mp 5	1 ⊢ (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜓)

Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1541   = wceq 1543  Ⅎwnf 1791   ∈ wcel 2112  Vcvv 3423

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2114  ax-12 2177

This theorem depends on definitions:  df-bi 210  df-an 400  df-3an 1091  df-ex 1788  df-nf 1792  df-clel 2818

This theorem is referenced by:  bj-ceqsalv  34981