Theorem bj-ceqsal 32076

Description: Remove from ceqsal 3205 dependency on ax-ext 2590 (and on df-cleq 2603, df-v 3175, df-clab 2597, df-sb 1868). (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 32074	. 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 195  ∀wal 1473   = wceq 1475  Ⅎwnf 1699   ∈ wcel 1977  Vcvv 3173

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  ax-7 1922  ax-12 2034

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-ex 1696  df-nf 1701  df-clel 2606

This theorem is referenced by:  bj-ceqsalv  32077