Mathbox for Glauco Siliprandi < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dfcleqf Structured version   Visualization version   GIF version

Theorem dfcleqf 40190
 Description: Equality connective between classes. Same as dfcleq 2771, 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 2963 1 (𝐴 = 𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 198  ∀wal 1599   = wceq 1601   ∈ wcel 2107  Ⅎwnfc 2919 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-ext 2754 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-tru 1605  df-ex 1824  df-nf 1828  df-cleq 2770  df-clel 2774  df-nfc 2921 This theorem is referenced by:  ssmapsn  40333  infnsuprnmpt  40380  preimagelt  41843  preimalegt  41844  pimrecltpos  41850  pimrecltneg  41864  smfaddlem1  41902  smflimsuplem7  41963
 Copyright terms: Public domain W3C validator