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Theorem bj-ceqsal 32076
Description: Remove from ceqsal 3205 dependency on ax-ext 2590 (and on df-cleq 2603, df-v 3175, df-clab 2597, df-sb 1868). (Contributed by BJ, 12-Oct-2019.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
bj-ceqsal.1 𝑥𝜓
bj-ceqsal.2 𝐴 ∈ V
bj-ceqsal.3 (𝑥 = 𝐴 → (𝜑𝜓))
Assertion
Ref Expression
bj-ceqsal (∀𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓)
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)

Proof of Theorem bj-ceqsal
StepHypRef Expression
1 bj-ceqsal.2 . 2 𝐴 ∈ V
2 bj-ceqsal.1 . . 3 𝑥𝜓
3 bj-ceqsal.3 . . 3 (𝑥 = 𝐴 → (𝜑𝜓))
42, 3bj-ceqsalgv 32074 . 2 (𝐴 ∈ V → (∀𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓))
51, 4ax-mp 5 1 (∀𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wal 1473   = wceq 1475  wnf 1699  wcel 1977  Vcvv 3173
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-ceqsalv  32077
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