Theorem eqnbrtrd 5118

Description: Substitution of equal classes into the negation of a binary relation. (Contributed by Glauco Siliprandi, 3-Jan-2021.)

Hypotheses

Ref	Expression
eqnbrtrd.1	⊢ (𝜑 → 𝐴 = 𝐵)
eqnbrtrd.2	⊢ (𝜑 → ¬ 𝐵𝑅𝐶)

Assertion

Ref	Expression
eqnbrtrd	⊢ (𝜑 → ¬ 𝐴𝑅𝐶)

Proof of Theorem eqnbrtrd

Step	Hyp	Ref	Expression
1		eqnbrtrd.2	. 2 ⊢ (𝜑 → ¬ 𝐵𝑅𝐶)
2		eqnbrtrd.1	. . 3 ⊢ (𝜑 → 𝐴 = 𝐵)
3	2	breq1d 5110	. 2 ⊢ (𝜑 → (𝐴𝑅𝐶 ↔ 𝐵𝑅𝐶))
4	1, 3	mtbird 325	1 ⊢ (𝜑 → ¬ 𝐴𝑅𝐶)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1542   class class class wbr 5100

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 3402  df-v 3444  df-dif 3906  df-un 3908  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101

This theorem is referenced by:  supgtoreq  9386  rlimno1  15589  pczndvds  16805  pcadd  16829  recld2  24774  itg2cnlem2  25734  dgrub  26210  gausslemma2dlem1a  27347  nosupbnd1lem1  27691  nosupbnd2lem1  27698  noinfbnd1lem1  27706  noinfbnd2  27714  mirbtwnhl  28768  mullt0b2d  42858  sqrtcval  44001