Theorem breqtrrdi 4153

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 4152	1 |- ( ph -> A R C )

Syntax hints:   -> wi 4   = wceq 1398   class class class wbr 4111

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 3217  df-sn 3697  df-pr 3698  df-op 3700  df-br 4112

This theorem is referenced by:  enpr2d  7066  fiunsnnn  7140  exmidpw2en  7174  unsnfi  7181  2omapfi  7273  eninl  7390  eninr  7391  difinfinf  7394  exmidfodomrlemr  7507  exmidfodomrlemrALT  7508  dju1en  7522  djucomen  7525  djuassen  7526  xpdjuen  7527  gtndiv  9679  intqfrac2  10688  uzenom  10794  xrmaxiflemval  11943  ege2le3  12365  eirraplem  12471  bitsfzo  12649  pcprendvds  12996  pcpremul  12999  pcfaclem  13055  infpnlem2  13066  2strstr1g  13356  lmcn2  15194  dveflem  15640  tangtx  15752  ioocosf1o  15768  lgsdirprm  15956  sbthom  16855  nconstwlpolemgt0  16899