Theorem eqnbrtrd 4036

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 4028	. 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 4018

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 2171

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-un 3148  df-sn 3613  df-pr 3614  df-op 3616  df-br 4019

This theorem is referenced by:  xnn0dcle  9834  xqltnle  10300  pczndvds  12351  pcadd  12375