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Theorem eqnbrtrd 5166
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 5158 . 2 (𝜑 → (𝐴𝑅𝐶𝐵𝑅𝐶))
41, 3mtbird 325 1 (𝜑 → ¬ 𝐴𝑅𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1537   class class class wbr 5148
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149
This theorem is referenced by:  supgtoreq  9508  rlimno1  15687  pczndvds  16899  pcadd  16923  recld2  24850  itg2cnlem2  25812  dgrub  26288  gausslemma2dlem1a  27424  nosupbnd1lem1  27768  nosupbnd2lem1  27775  noinfbnd1lem1  27783  noinfbnd2  27791  mirbtwnhl  28703  sqrtcval  43631
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