Theorem bj-ceqsalg 37413

Description: Remove from ceqsalg 3498 dependency on ax-ext 2741 (and on df-cleq 2761 and df-v 3465). See also bj-ceqsalgv 37415. (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 2851	. 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 37412	. 2 ⊢ (∃𝑥 𝑥 = 𝐴 → (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜓))
5	1, 4	syl 18	1 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜓))

Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1565   = wceq 1567  ∃wex 1806  Ⅎwnf 1810   ∈ wcel 2149

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-12 2219

This theorem depends on definitions:  df-bi 210  df-an 401  df-3an 1103  df-tru 1570  df-ex 1807  df-nf 1811  df-sb 2098  df-clab 2748  df-clel 2844

This theorem is referenced by: (None)