Theorem bj-ceqsalv 35774

Description: Remove from ceqsalv 3512 dependency on ax-ext 2704 (and on df-cleq 2725, df-v 3477, df-clab 2711, 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 1918	. 2 ⊢ Ⅎ𝑥𝜓
2		bj-ceqsalv.1	. 2 ⊢ 𝐴 ∈ V
3		bj-ceqsalv.2	. 2 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
4	1, 2, 3	bj-ceqsal 35773	1 ⊢ (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜓)

Syntax hints:   → wi 4   ↔ wb 205  ∀wal 1540   = wceq 1542   ∈ wcel 2107  Vcvv 3475

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-12 2172

This theorem depends on definitions:  df-bi 206  df-an 398  df-3an 1090  df-ex 1783  df-nf 1787  df-clel 2811

This theorem is referenced by: (None)