Theorem breqtrrdi 4170

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 2242	. 2 |- B = C
4	1, 3	breqtrdi 4169	1 |- ( ph -> A R C )

Syntax hints:   -> wi 4   = wceq 1402   class class class wbr 4128

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224  df-sn 3714  df-pr 3715  df-op 3717  df-br 4129

This theorem is referenced by:  enpr2d  7105  fiunsnnn  7179  exmidpw2en  7213  unsnfi  7220  2omapfi  7314  eninl  7431  eninr  7432  difinfinf  7435  exmidfodomrlemr  7548  exmidfodomrlemrALT  7549  dju1en  7563  djucomen  7566  djuassen  7567  xpdjuen  7568  gtndiv  9724  intqfrac2  10739  uzenom  10845  xrmaxiflemval  11999  ege2le3  12421  eirraplem  12527  bitsfzo  12705  pcprendvds  13052  pcpremul  13055  pcfaclem  13111  infpnlem2  13122  2strstr1g  13459  lmcn2  15364  dveflem  15810  tangtx  15922  ioocosf1o  15938  lgsdirprm  16136  sbthom  17045  nconstwlpolemgt0  17088