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Theorem brneqtrd 41727
 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 5042 . 2 (𝜑 → (𝐴𝑅𝐵𝐴𝑅𝐶))
41, 3mtbid 327 1 (𝜑 → ¬ 𝐴𝑅𝐶)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   = wceq 1538   class class class wbr 5030 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-v 3443  df-un 3886  df-sn 4526  df-pr 4528  df-op 4532  df-br 5031 This theorem is referenced by: (None)
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