Theorem brneqtrd 45748

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	⊢ (𝜑 → 𝐵 = 𝐶)

Assertion

Ref	Expression
brneqtrd	⊢ (𝜑 → ¬ 𝐴𝑅𝐶)

Proof of Theorem brneqtrd

Step	Hyp	Ref	Expression
1		brneqtrd.1	. 2 ⊢ (𝜑 → ¬ 𝐴𝑅𝐵)
2		brneqtrd.2	. . 3 ⊢ (𝜑 → 𝐵 = 𝐶)
3	2	breq2d 5126	. 2 ⊢ (𝜑 → (𝐴𝑅𝐵 ↔ 𝐴𝑅𝐶))
4	1, 3	mtbid 327	1 ⊢ (𝜑 → ¬ 𝐴𝑅𝐶)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1568   class class class wbr 5114

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2152  ax-9 2160  ax-ext 2742

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-sb 2099  df-clab 2749  df-cleq 2762  df-clel 2845  df-rab 3424  df-v 3464  df-dif 3916  df-un 3918  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5115

This theorem is referenced by:  rexanuz2nf  46158