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Theorem brneqtrd 38770
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 4635 . 2 (𝜑 → (𝐴𝑅𝐵𝐴𝑅𝐶))
41, 3mtbid 314 1 (𝜑 → ¬ 𝐴𝑅𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1480   class class class wbr 4623
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-rab 2917  df-v 3192  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-br 4624
This theorem is referenced by: (None)
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