Theorem breqtrrdi 4043

Description: A chained equality inference for a binary relation. (Contributed by NM, 24-Apr-2005.)

Hypotheses

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

Assertion

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

Proof of Theorem breqtrrdi

Step	Hyp	Ref	Expression
1		breqtrrdi.1	. 2 |- ( ph -> A R B )
2		breqtrrdi.2	. . 3 |- C = B
3	2	eqcomi 2181	. 2 |- B = C
4	1, 3	breqtrdi 4042	1 |- ( ph -> A R C )

Syntax hints:   -> wi 4   = wceq 1353   class class class wbr 4001

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-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-un 3133  df-sn 3598  df-pr 3599  df-op 3601  df-br 4002

This theorem is referenced by:  enpr2d  6812  fiunsnnn  6876  unsnfi  6913  eninl  7091  eninr  7092  difinfinf  7095  exmidfodomrlemr  7196  exmidfodomrlemrALT  7197  dju1en  7207  djucomen  7210  djuassen  7211  xpdjuen  7212  gtndiv  9342  intqfrac2  10312  uzenom  10418  xrmaxiflemval  11249  ege2le3  11670  eirraplem  11775  pcprendvds  12280  pcpremul  12283  pcfaclem  12337  infpnlem2  12348  2strstr1g  12570  lmcn2  13562  dveflem  13969  tangtx  14041  ioocosf1o  14057  lgsdirprm  14217  sbthom  14545  nconstwlpolemgt0  14582