Theorem eqbrtrd 3784

Description: Substitution of equal classes into a binary relation. (Contributed by NM, 8-Oct-1999.)

Hypotheses

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

Assertion

Ref	Expression
eqbrtrd	|- ( ph -> A R C )

Proof of Theorem eqbrtrd

Step	Hyp	Ref	Expression
1		eqbrtrd.2	. 2 |- ( ph -> B R C )
2		eqbrtrd.1	. . 3 |- ( ph -> A = B )
3	2	breq1d 3774	. 2 |- ( ph -> ( A R C <-> B R C ) )
4	1, 3	mpbird 156	1 |- ( ph -> A R C )

Syntax hints:   -> wi 4   = wceq 1243   class class class wbr 3764

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765

This theorem is referenced by:  eqbrtrrd  3786  dif1en  6337  prarloclemcalc  6600  ltexprlemopu  6701  recexprlemloc  6729  caucvgprprlemloccalc  6782  divge1  8649  xltnegi  8748  ubmelm1fzo  9082  qbtwnrelemcalc  9110  ceiqm1l  9153  ceilqm1lt  9154  ceilqle  9156  modqlt  9175  bernneq  9369  resqrexlemdec  9609  resqrexlemcalc2  9613  resqrexlemglsq  9620  resqrexlemga  9621  abslt  9684  amgm2  9714  icodiamlt  9776  climconst  9811  iserclim0  9826  mulcn2  9833  iiserex  9859  climlec2  9861  iserige0  9863  climcau  9866  climcvg1nlem  9868  qdencn  10124