Theorem eqnbrtrd 4008

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 4000	. 2 |- ( ph -> ( A R C <-> B R C ) )
4	1, 3	mtbird 669	1 |- ( ph -> -. A R C )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1349   class class class wbr 3990

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 610  ax-in2 611  ax-io 705  ax-5 1441  ax-7 1442  ax-gen 1443  ax-ie1 1487  ax-ie2 1488  ax-8 1498  ax-10 1499  ax-11 1500  ax-i12 1501  ax-bndl 1503  ax-4 1504  ax-17 1520  ax-i9 1524  ax-ial 1528  ax-i5r 1529  ax-ext 2153

This theorem depends on definitions:  df-bi 116  df-3an 976  df-tru 1352  df-nf 1455  df-sb 1757  df-clab 2158  df-cleq 2164  df-clel 2167  df-nfc 2302  df-v 2733  df-un 3126  df-sn 3590  df-pr 3591  df-op 3593  df-br 3991

This theorem is referenced by:  xnn0dcle  9763  pczndvds  12273  pcadd  12297