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Theorem brneqtrd 44365
Description: Substitution of equal classes into the negation of a binary relation. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
Hypotheses
Ref Expression
brneqtrd.1 (𝜑 → ¬ 𝐴𝑅𝐵)
brneqtrd.2 (𝜑𝐵 = 𝐶)
Assertion
Ref Expression
brneqtrd (𝜑 → ¬ 𝐴𝑅𝐶)

Proof of Theorem brneqtrd
StepHypRef Expression
1 brneqtrd.1 . 2 (𝜑 → ¬ 𝐴𝑅𝐵)
2 brneqtrd.2 . . 3 (𝜑𝐵 = 𝐶)
32breq2d 5154 . 2 (𝜑 → (𝐴𝑅𝐵𝐴𝑅𝐶))
41, 3mtbid 324 1 (𝜑 → ¬ 𝐴𝑅𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1534   class class class wbr 5142
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2698
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2705  df-cleq 2719  df-clel 2805  df-rab 3428  df-v 3471  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-br 5143
This theorem is referenced by:  rexanuz2nf  44798
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