Theorem eqnbrtrd 4051

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

Hypotheses

Ref	Expression
eqnbrtrd.1	|- ( ph -> A = B )
eqnbrtrd.2	|- ( ph -> -. B R C )

Assertion

Ref	Expression
eqnbrtrd	|- ( ph -> -. A R C )

Proof of Theorem eqnbrtrd

Step	Hyp	Ref	Expression
1		eqnbrtrd.2	. 2 |- ( ph -> -. B R C )
2		eqnbrtrd.1	. . 3 |- ( ph -> A = B )
3	2	breq1d 4043	. 2 |- ( ph -> ( A R C <-> B R C ) )
4	1, 3	mtbird 674	1 |- ( ph -> -. A R C )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1364   class class class wbr 4033

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-un 3161  df-sn 3628  df-pr 3629  df-op 3631  df-br 4034

This theorem is referenced by:  xnn0dcle  9874  xqltnle  10342  pczndvds  12461  pcadd  12485  gausslemma2dlem1a  15266