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Theorem bj-ceqsalg 32072
Description: Remove from ceqsalg 3203 dependency on ax-ext 2590 (and on df-cleq 2603 and df-v 3175). See also bj-ceqsalgv 32074. (Contributed by BJ, 12-Oct-2019.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
bj-ceqsalg.1 𝑥𝜓
bj-ceqsalg.2 (𝑥 = 𝐴 → (𝜑𝜓))
Assertion
Ref Expression
bj-ceqsalg (𝐴𝑉 → (∀𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓))
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)   𝑉(𝑥)

Proof of Theorem bj-ceqsalg
StepHypRef Expression
1 bj-elisset 32056 . 2 (𝐴𝑉 → ∃𝑥 𝑥 = 𝐴)
2 bj-ceqsalg.1 . . 3 𝑥𝜓
3 bj-ceqsalg.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  wnf 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-sb 1868  df-clab 2597  df-clel 2606
This theorem is referenced by: (None)
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