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Theorem bj-ceqsalgvALT 33227
Description: Alternate proof of bj-ceqsalgv 33226. (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 1873 . 2 𝑥(𝑥 = 𝐴 → (𝜑𝜓))
4 bj-ceqsaltv 33222 . 2 ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ 𝐴𝑉) → (∀𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓))
51, 3, 4mp3an12 1565 1 (𝐴𝑉 → (∀𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wal 1632   = wceq 1634  wnf 1859  wcel 2148
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1873  ax-4 1888  ax-5 1994  ax-6 2060  ax-7 2096  ax-12 2206
This theorem depends on definitions:  df-bi 198  df-an 384  df-or 864  df-3an 1100  df-ex 1856  df-nf 1861  df-clel 2770
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator