Theorem breqtrrdi 4060

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 2193	. 2 |- B = C
4	1, 3	breqtrdi 4059	1 |- ( ph -> A R C )

Syntax hints:   -> wi 4   = wceq 1364   class class class wbr 4018

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-un 3148  df-sn 3613  df-pr 3614  df-op 3616  df-br 4019

This theorem is referenced by:  enpr2d  6844  fiunsnnn  6910  exmidpw2en  6941  unsnfi  6948  eninl  7127  eninr  7128  difinfinf  7131  exmidfodomrlemr  7232  exmidfodomrlemrALT  7233  dju1en  7243  djucomen  7246  djuassen  7247  xpdjuen  7248  gtndiv  9379  intqfrac2  10352  uzenom  10458  xrmaxiflemval  11293  ege2le3  11714  eirraplem  11819  pcprendvds  12325  pcpremul  12328  pcfaclem  12384  infpnlem2  12395  2strstr1g  12636  lmcn2  14257  dveflem  14664  tangtx  14736  ioocosf1o  14752  lgsdirprm  14913  sbthom  15253  nconstwlpolemgt0  15291