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Theorem bj-ceqsalg 36276
Description: Remove from ceqsalg 3502 dependency on ax-ext 2697 (and on df-cleq 2718 and df-v 3470). See also bj-ceqsalgv 36278. (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 elisset 2809 . 2 (𝐴𝑉 → ∃𝑥 𝑥 = 𝐴)
2 bj-ceqsalg.1 . . 3 𝑥𝜓
3 bj-ceqsalg.2 . . 3 (𝑥 = 𝐴 → (𝜑𝜓))
42, 3bj-ceqsalg0 36275 . 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 205  wal 1531   = wceq 1533  wex 1773  wnf 1777  wcel 2098
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-12 2163
This theorem depends on definitions:  df-bi 206  df-an 396  df-3an 1086  df-tru 1536  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2704  df-clel 2804
This theorem is referenced by: (None)
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