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Theorem bj-ceqsalgALT 34604
 Description: Alternate proof of bj-ceqsalg 34603. (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 1798 . 2 𝑥(𝑥 = 𝐴 → (𝜑𝜓))
4 bj-ceqsalt 34600 . 2 ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ 𝐴𝑉) → (∀𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓))
51, 3, 4mp3an12 1449 1 (𝐴𝑉 → (∀𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1537   = wceq 1539  Ⅎwnf 1786   ∈ wcel 2112 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-12 2176 This theorem depends on definitions:  df-bi 210  df-an 401  df-3an 1087  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2071  df-clab 2737  df-clel 2831 This theorem is referenced by: (None)
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