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Theorem bj-abeq1 33225
Description: Remove dependency on ax-13 2352 from abeq1 2876. Remark: the theorems abeq2i 2878, abeq1i 2879, abeq2d 2877 do not use ax-11 2198 or ax-13 2352. (Contributed by BJ, 23-Jun-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-abeq1 ({𝑥𝜑} = 𝐴 ↔ ∀𝑥(𝜑𝑥𝐴))
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem bj-abeq1
StepHypRef Expression
1 bj-abeq2 33224 . 2 (𝐴 = {𝑥𝜑} ↔ ∀𝑥(𝑥𝐴𝜑))
2 eqcom 2772 . 2 ({𝑥𝜑} = 𝐴𝐴 = {𝑥𝜑})
3 bicom 213 . . 3 ((𝜑𝑥𝐴) ↔ (𝑥𝐴𝜑))
43albii 1914 . 2 (∀𝑥(𝜑𝑥𝐴) ↔ ∀𝑥(𝑥𝐴𝜑))
51, 2, 43bitr4i 294 1 ({𝑥𝜑} = 𝐴 ↔ ∀𝑥(𝜑𝑥𝐴))
Colors of variables: wff setvar class
Syntax hints:  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 2070  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 2063  df-clab 2752  df-cleq 2758  df-clel 2761
This theorem is referenced by: (None)
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