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Theorem brneqtrd 43374
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 5118 . 2 (𝜑 → (𝐴𝑅𝐵𝐴𝑅𝐶))
41, 3mtbid 324 1 (𝜑 → ¬ 𝐴𝑅𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1542   class class class wbr 5106
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-br 5107
This theorem is referenced by:  rexanuz2nf  43814
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