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Theorem dfcleqf 39776
Description: Equality connective between classes. Same as dfcleq 2765, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
Hypotheses
Ref Expression
dfcleqf.1 𝑥𝐴
dfcleqf.2 𝑥𝐵
Assertion
Ref Expression
dfcleqf (𝐴 = 𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))

Proof of Theorem dfcleqf
StepHypRef Expression
1 dfcleqf.1 . 2 𝑥𝐴
2 dfcleqf.2 . 2 𝑥𝐵
31, 2cleqf 2939 1 (𝐴 = 𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wal 1629   = wceq 1631  wcel 2145  wnfc 2900
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-cleq 2764  df-clel 2767  df-nfc 2902
This theorem is referenced by:  ssmapsn  39925  infnsuprnmpt  39980  preimagelt  41429  preimalegt  41430  pimrecltpos  41436  pimrecltneg  41450  smfaddlem1  41488  smflimsuplem7  41549
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