Theorem eqnbrtrd 4101

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 4093	. 2 |- ( ph -> ( A R C <-> B R C ) )
4	1, 3	mtbird 677	1 |- ( ph -> -. A R C )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1395   class class class wbr 4083

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-un 3201  df-sn 3672  df-pr 3673  df-op 3675  df-br 4084

This theorem is referenced by:  xnn0dcle  9998  xqltnle  10487  pczndvds  12839  pcadd  12863  gausslemma2dlem1a  15737