Theorem brneqtrd 45388

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 5111	. 2 ⊢ (𝜑 → (𝐴𝑅𝐵 ↔ 𝐴𝑅𝐶))
4	1, 3	mtbid 324	1 ⊢ (𝜑 → ¬ 𝐴𝑅𝐶)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1542   class class class wbr 5099

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-rab 3401  df-v 3443  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4287  df-if 4481  df-sn 4582  df-pr 4584  df-op 4588  df-br 5100

This theorem is referenced by:  rexanuz2nf  45803