Theorem eqnbrtrd 4063

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 4055	. 2 |- ( ph -> ( A R C <-> B R C ) )
4	1, 3	mtbird 675	1 |- ( ph -> -. A R C )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1373   class class class wbr 4045

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-un 3170  df-sn 3639  df-pr 3640  df-op 3642  df-br 4046

This theorem is referenced by:  xnn0dcle  9926  xqltnle  10412  pczndvds  12672  pcadd  12696  gausslemma2dlem1a  15568