Theorem eqnbrtrd 4047

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 4039	. 2 ⊢ (𝜑 → (𝐴𝑅𝐶 ↔ 𝐵𝑅𝐶))
4	1, 3	mtbird 674	1 ⊢ (𝜑 → ¬ 𝐴𝑅𝐶)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1364   class class class wbr 4029

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-un 3157  df-sn 3624  df-pr 3625  df-op 3627  df-br 4030

This theorem is referenced by:  xnn0dcle  9868  xqltnle  10336  pczndvds  12454  pcadd  12478  gausslemma2dlem1a  15174