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Theorem eqnbrtrd 5170
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 5162 . 2 (𝜑 → (𝐴𝑅𝐶𝐵𝑅𝐶))
41, 3mtbird 324 1 (𝜑 → ¬ 𝐴𝑅𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1533   class class class wbr 5152
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-br 5153
This theorem is referenced by:  supgtoreq  9503  rlimno1  15642  pczndvds  16843  pcadd  16867  recld2  24758  itg2cnlem2  25720  dgrub  26196  gausslemma2dlem1a  27326  nosupbnd1lem1  27669  nosupbnd2lem1  27676  noinfbnd1lem1  27684  noinfbnd2  27692  mirbtwnhl  28512  sqrtcval  43120
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