Theorem brneqtrd 43408

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 5122	. 2 ⊢ (𝜑 → (𝐴𝑅𝐵 ↔ 𝐴𝑅𝐶))
4	1, 3	mtbid 323	1 ⊢ (𝜑 → ¬ 𝐴𝑅𝐶)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1541   class class class wbr 5110

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-rab 3406  df-v 3448  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-br 5111

This theorem is referenced by:  rexanuz2nf  43848