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Theorem bj-ceqsalgALT 32854
Description: Alternate proof of bj-ceqsalg 32853. (Contributed by BJ, 12-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
bj-ceqsalg.1 𝑥𝜓
bj-ceqsalg.2 (𝑥 = 𝐴 → (𝜑𝜓))
Assertion
Ref Expression
bj-ceqsalgALT (𝐴𝑉 → (∀𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓))
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)   𝑉(𝑥)

Proof of Theorem bj-ceqsalgALT
StepHypRef Expression
1 bj-ceqsalg.1 . 2 𝑥𝜓
2 bj-ceqsalg.2 . . 3 (𝑥 = 𝐴 → (𝜑𝜓))
32ax-gen 1720 . 2 𝑥(𝑥 = 𝐴 → (𝜑𝜓))
4 bj-ceqsalt 32850 . 2 ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ 𝐴𝑉) → (∀𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓))
51, 3, 4mp3an12 1412 1 (𝐴𝑉 → (∀𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wal 1479   = wceq 1481  wnf 1706  wcel 1988
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-12 2045
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-clel 2616
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator