Theorem eqnbrtrd 5160

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 5152	. 2 ⊢ (𝜑 → (𝐴𝑅𝐶 ↔ 𝐵𝑅𝐶))
4	1, 3	mtbird 325	1 ⊢ (𝜑 → ¬ 𝐴𝑅𝐶)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1539   class class class wbr 5142

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-br 5143

This theorem is referenced by:  supgtoreq  9511  rlimno1  15691  pczndvds  16904  pcadd  16928  recld2  24837  itg2cnlem2  25798  dgrub  26274  gausslemma2dlem1a  27410  nosupbnd1lem1  27754  nosupbnd2lem1  27761  noinfbnd1lem1  27769  noinfbnd2  27777  mirbtwnhl  28689  sqrtcval  43659