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Theorem bj-ceqsalv 32077
 Description: Remove from ceqsalv 3206 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-ceqsalv.1 𝐴 ∈ V
bj-ceqsalv.2 (𝑥 = 𝐴 → (𝜑𝜓))
Assertion
Ref Expression
bj-ceqsalv (∀𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓)
Distinct variable groups:   𝑥,𝐴   𝜓,𝑥
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem bj-ceqsalv
StepHypRef Expression
1 nfv 1830 . 2 𝑥𝜓
2 bj-ceqsalv.1 . 2 𝐴 ∈ V
3 bj-ceqsalv.2 . 2 (𝑥 = 𝐴 → (𝜑𝜓))
41, 2, 3bj-ceqsal 32076 1 (∀𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195  ∀wal 1473   = wceq 1475   ∈ 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: (None)
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