Theorem brneqtrd 45539

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 5087	. 2 ⊢ (𝜑 → (𝐴𝑅𝐵 ↔ 𝐴𝑅𝐶))
4	1, 3	mtbid 326	1 ⊢ (𝜑 → ¬ 𝐴𝑅𝐶)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1548   class class class wbr 5075

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-rab 3394  df-v 3435  df-dif 3888  df-un 3890  df-ss 3902  df-nul 4265  df-if 4458  df-sn 4559  df-pr 4561  df-op 4565  df-br 5076

This theorem is referenced by:  rexanuz2nf  45949