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Theorem dfcleqf 38707
Description: Equality connective between classes. Same as dfcleq 2620, 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 2792 1 (𝐴 = 𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wal 1478   = wceq 1480  wcel 1992  wnfc 2754
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-cleq 2619  df-clel 2622  df-nfc 2756
This theorem is referenced by:  ssmapsn  38849  infnsuprnmpt  38909  preimagelt  40187  preimalegt  40188  pimrecltpos  40194  pimrecltneg  40208  smfaddlem1  40246  smflimsuplem7  40307
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