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Theorem bj-abbi2dv 31968
Description: Remove dependency on ax-13 2234 from abbi2dv 2729. (Contributed by BJ, 23-Jun-2019.) (Proof modification is discouraged.)
Hypothesis
Ref Expression
bj-abbi2dv.1 (𝜑 → (𝑥𝐴𝜓))
Assertion
Ref Expression
bj-abbi2dv (𝜑𝐴 = {𝑥𝜓})
Distinct variable groups:   𝑥,𝐴   𝜑,𝑥
Allowed substitution hint:   𝜓(𝑥)

Proof of Theorem bj-abbi2dv
StepHypRef Expression
1 bj-abbi2dv.1 . . 3 (𝜑 → (𝑥𝐴𝜓))
21alrimiv 1842 . 2 (𝜑 → ∀𝑥(𝑥𝐴𝜓))
3 bj-abeq2 31961 . 2 (𝐴 = {𝑥𝜓} ↔ ∀𝑥(𝑥𝐴𝜓))
42, 3sylibr 223 1 (𝜑𝐴 = {𝑥𝜓})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wal 1473   = wceq 1475  wcel 1977  {cab 2596
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-10 2006  ax-11 2021  ax-12 2034  ax-ext 2590
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606
This theorem is referenced by:  bj-abbi1dv  31969  bj-sbab  31972
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