Theorem breqtrrdi 4156

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 2238	. 2 |- B = C
4	1, 3	breqtrdi 4155	1 |- ( ph -> A R C )

Syntax hints:   -> wi 4   = wceq 1398   class class class wbr 4114

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-v 2817  df-un 3218  df-sn 3700  df-pr 3701  df-op 3703  df-br 4115

This theorem is referenced by:  enpr2d  7077  fiunsnnn  7151  exmidpw2en  7185  unsnfi  7192  2omapfi  7284  eninl  7401  eninr  7402  difinfinf  7405  exmidfodomrlemr  7518  exmidfodomrlemrALT  7519  dju1en  7533  djucomen  7536  djuassen  7537  xpdjuen  7538  gtndiv  9694  intqfrac2  10708  uzenom  10814  xrmaxiflemval  11963  ege2le3  12385  eirraplem  12491  bitsfzo  12669  pcprendvds  13016  pcpremul  13019  pcfaclem  13075  infpnlem2  13086  2strstr1g  13422  lmcn2  15274  dveflem  15720  tangtx  15832  ioocosf1o  15848  lgsdirprm  16036  sbthom  16945  nconstwlpolemgt0  16989