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Theorem eqnbrtrd 4948
 Description: Substitution of equal classes into the negation of a binary relation. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
Hypotheses
Ref Expression
eqnbrtrd.1 (𝜑𝐴 = 𝐵)
eqnbrtrd.2 (𝜑 → ¬ 𝐵𝑅𝐶)
Assertion
Ref Expression
eqnbrtrd (𝜑 → ¬ 𝐴𝑅𝐶)

Proof of Theorem eqnbrtrd
StepHypRef Expression
1 eqnbrtrd.2 . 2 (𝜑 → ¬ 𝐵𝑅𝐶)
2 eqnbrtrd.1 . . 3 (𝜑𝐴 = 𝐵)
32breq1d 4940 . 2 (𝜑 → (𝐴𝑅𝐶𝐵𝑅𝐶))
41, 3mtbird 317 1 (𝜑 → ¬ 𝐴𝑅𝐶)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   = wceq 1507   class class class wbr 4930 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-ext 2750 This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-rab 3097  df-v 3417  df-dif 3834  df-un 3836  df-in 3838  df-ss 3845  df-nul 4181  df-if 4352  df-sn 4443  df-pr 4445  df-op 4449  df-br 4931 This theorem is referenced by:  supgtoreq  8731  rlimno1  14874  pczndvds  16060  pcadd  16084  recld2  23128  itg2cnlem2  24069  dgrub  24530  gausslemma2dlem1a  25646  mirbtwnhl  26171  noprefixmo  32723  nosupbnd1lem1  32729  nosupbnd2lem1  32736
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