Theorem eqnbrtrd 5084

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 5076	. 2 ⊢ (𝜑 → (𝐴𝑅𝐶 ↔ 𝐵𝑅𝐶))
4	1, 3	mtbird 327	1 ⊢ (𝜑 → ¬ 𝐴𝑅𝐶)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1537   class class class wbr 5066

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-rab 3147  df-v 3496  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-br 5067

This theorem is referenced by:  supgtoreq  8934  rlimno1  15010  pczndvds  16201  pcadd  16225  recld2  23422  itg2cnlem2  24363  dgrub  24824  gausslemma2dlem1a  25941  mirbtwnhl  26466  noprefixmo  33202  nosupbnd1lem1  33208  nosupbnd2lem1  33215