Theorem bj-ceqsal 36906

Description: Remove from ceqsal 3472 dependency on ax-ext 2702 (and on df-cleq 2722, df-v 3436, df-clab 2709, df-sb 2067). (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 36904	. 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 1539   = wceq 1541  Ⅎwnf 1784   ∈ wcel 2110  Vcvv 3434

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2112  ax-12 2179

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-ex 1781  df-nf 1785  df-clel 2804

This theorem is referenced by:  bj-ceqsalv  36907