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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 𝑥𝜓
bj-ceqsalgv.2 (𝑥 = 𝐴 → (𝜑𝜓))
Assertion
Ref Expression
bj-ceqsalgv (𝐴𝑉 → (∀𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓))
Distinct variable groups:   𝑥,𝐴   𝑥,𝑉
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)

Proof of Theorem bj-ceqsalgv
StepHypRef Expression
1 bj-elissetv 32055 . 2 (𝐴𝑉 → ∃𝑥 𝑥 = 𝐴)
2 bj-ceqsalgv.1 . . 3 𝑥𝜓
3 bj-ceqsalgv.2 . . 3 (𝑥 = 𝐴 → (𝜑𝜓))
42, 3bj-ceqsalg0 32071 . 2 (∃𝑥 𝑥 = 𝐴 → (∀𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓))
51, 4syl 17 1 (𝐴𝑉 → (∀𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓))
 Colors of variables: wff setvar class 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
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