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Theorem bj-ceqsalgvALT 32075
 Description: Alternate proof of bj-ceqsalgv 32074. (Contributed by BJ, 12-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
bj-ceqsalgv.1 𝑥𝜓
bj-ceqsalgv.2 (𝑥 = 𝐴 → (𝜑𝜓))
Assertion
Ref Expression
bj-ceqsalgvALT (𝐴𝑉 → (∀𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓))
Distinct variable groups:   𝑥,𝐴   𝑥,𝑉
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)

Proof of Theorem bj-ceqsalgvALT
StepHypRef Expression
1 bj-ceqsalgv.1 . 2 𝑥𝜓
2 bj-ceqsalgv.2 . . 3 (𝑥 = 𝐴 → (𝜑𝜓))
32ax-gen 1713 . 2 𝑥(𝑥 = 𝐴 → (𝜑𝜓))
4 bj-ceqsaltv 32070 . 2 ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ 𝐴𝑉) → (∀𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓))
51, 3, 4mp3an12 1406 1 (𝐴𝑉 → (∀𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195  ∀wal 1473   = wceq 1475  Ⅎwnf 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-clel 2606 This theorem is referenced by: (None)
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