Theorem bj-ceqsalg 36868

Description: Remove from ceqsalg 3516 dependency on ax-ext 2707 (and on df-cleq 2728 and df-v 3481). See also bj-ceqsalgv 36870. (Contributed by BJ, 12-Oct-2019.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
bj-ceqsalg.1	⊢ Ⅎ𝑥𝜓
bj-ceqsalg.2	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
bj-ceqsalg	⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜓))

Distinct variable group:   𝑥,𝐴

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)   𝑉(𝑥)

Proof of Theorem bj-ceqsalg

Step	Hyp	Ref	Expression
1		elisset 2822	. 2 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴)
2		bj-ceqsalg.1	. . 3 ⊢ Ⅎ𝑥𝜓
3		bj-ceqsalg.2	. . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
4	2, 3	bj-ceqsalg0 36867	. 2 ⊢ (∃𝑥 𝑥 = 𝐴 → (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜓))
5	1, 4	syl 17	1 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜓))

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1538   = wceq 1540  ∃wex 1779  Ⅎwnf 1783   ∈ wcel 2108

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 1967  ax-7 2007  ax-8 2110  ax-12 2177

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1089  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2714  df-clel 2815

This theorem is referenced by: (None)