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Theorem dfcleqf 39569
 Description: Equality connective between classes. Same as dfcleq 2645, 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 2819 1 (𝐴 = 𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 196  ∀wal 1521   = wceq 1523   ∈ wcel 2030  Ⅎwnfc 2780 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-cleq 2644  df-clel 2647  df-nfc 2782 This theorem is referenced by:  ssmapsn  39722  infnsuprnmpt  39779  preimagelt  41233  preimalegt  41234  pimrecltpos  41240  pimrecltneg  41254  smfaddlem1  41292  smflimsuplem7  41353
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