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Theorem brneqtrd 40765
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 4941 . 2 (𝜑 → (𝐴𝑅𝐵𝐴𝑅𝐶))
41, 3mtbid 316 1 (𝜑 → ¬ 𝐴𝑅𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1507   class class class wbr 4929
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 3832  df-un 3834  df-in 3836  df-ss 3843  df-nul 4179  df-if 4351  df-sn 4442  df-pr 4444  df-op 4448  df-br 4930
This theorem is referenced by: (None)
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