Theorem bj-ceqsalgv 32074

Description: Version of bj-ceqsalg 32072 with a dv condition on 𝑥, 𝑉, removing dependency on df-sb 1868 and df-clab 2597. Prefer its use over bj-ceqsalg 32072 when sufficient (in particular when 𝑉 is substituted for V). (Contributed by BJ, 12-Oct-2019.) (Proof modification is discouraged.)

Hypotheses

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

Assertion

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

Distinct variable groups:   𝑥,𝐴   𝑥,𝑉

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

Proof of Theorem bj-ceqsalgv

Step	Hyp	Ref	Expression
1		bj-elissetv 32055	. 2 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴)
2		bj-ceqsalgv.1	. . 3 ⊢ Ⅎ𝑥𝜓
3		bj-ceqsalgv.2	. . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
4	2, 3	bj-ceqsalg0 32071	. 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 195  ∀wal 1473   = wceq 1475  ∃wex 1695  Ⅎwnf 1699   ∈ wcel 1977

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-12 2034

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-ex 1696  df-nf 1701  df-clel 2606

This theorem is referenced by:  bj-ceqsal  32076