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Theorem bj-abbi2dv 33142
 Description: Remove dependency on ax-13 2352 from abbi2dv 2885. (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 2022 . 2 (𝜑 → ∀𝑥(𝑥𝐴𝜓))
3 bj-abeq2 33135 . 2 (𝐴 = {𝑥𝜓} ↔ ∀𝑥(𝑥𝐴𝜓))
42, 3sylibr 225 1 (𝜑𝐴 = {𝑥𝜓})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 197  ∀wal 1650   = wceq 1652   ∈ wcel 2155  {cab 2751 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-ext 2743 This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-clab 2752  df-cleq 2758  df-clel 2761 This theorem is referenced by:  bj-abbi1dv  33143  bj-sbab  33146
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