Theorem bj-ceqsalv 34234

Description: Remove from ceqsalv 3529 dependency on ax-ext 2792 (and on df-cleq 2813, df-v 3493, df-clab 2799, df-sb 2069). (Contributed by BJ, 12-Oct-2019.) (Proof modification is discouraged.)

Hypotheses

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

Assertion

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

Distinct variable groups:   𝑥,𝐴   𝜓,𝑥

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem bj-ceqsalv

Step	Hyp	Ref	Expression
1		nfv 1914	. 2 ⊢ Ⅎ𝑥𝜓
2		bj-ceqsalv.1	. 2 ⊢ 𝐴 ∈ V
3		bj-ceqsalv.2	. 2 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
4	1, 2, 3	bj-ceqsal 34233	1 ⊢ (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜓)

Syntax hints:   → wi 4   ↔ wb 208  ∀wal 1534   = wceq 1536   ∈ wcel 2113  Vcvv 3491

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 1084  df-ex 1780  df-nf 1784  df-clel 2892

This theorem is referenced by: (None)