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Theorem bj-ceqsalg 34102
Description: Remove from ceqsalg 3527 dependency on ax-ext 2790 (and on df-cleq 2811 and df-v 3494). See also bj-ceqsalgv 34104. (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 bj-elisset 34089 . 2 (𝐴𝑉 → ∃𝑥 𝑥 = 𝐴)
2 bj-ceqsalg.1 . . 3 𝑥𝜓
3 bj-ceqsalg.2 . . 3 (𝑥 = 𝐴 → (𝜑𝜓))
42, 3bj-ceqsalg0 34101 . 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 207  wal 1526   = wceq 1528  wex 1771  wnf 1775  wcel 2105
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-12 2167
This theorem depends on definitions:  df-bi 208  df-an 397  df-3an 1081  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-clel 2890
This theorem is referenced by: (None)
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