Theorem bj-ceqsal 34213

Description: Remove from ceqsal 3534 dependency on ax-ext 2796 (and on df-cleq 2817, df-v 3499, df-clab 2803, df-sb 2069). (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 34211	. 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 208  ∀wal 1534   = wceq 1536  Ⅎwnf 1783   ∈ wcel 2113  Vcvv 3497

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-12 2176

This theorem depends on definitions:  df-bi 209  df-an 399  df-3an 1085  df-ex 1780  df-nf 1784  df-clel 2896

This theorem is referenced by:  bj-ceqsalv  34214