Theorem bj-ceqsal 36888

Description: Remove from ceqsal 3518 dependency on ax-ext 2707 (and on df-cleq 2728, df-v 3481, df-clab 2714, df-sb 2064). (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 36886	. 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 206  ∀wal 1536   = wceq 1538  Ⅎwnf 1781   ∈ wcel 2107  Vcvv 3479

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-12 2176

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-ex 1778  df-nf 1782  df-clel 2815

This theorem is referenced by:  bj-ceqsalv  36889