Theorem breqtrrdi 4172

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 4171	1 |- ( ph -> A R C )

This proof depends on syntax axioms:   -> wi 4   = wceq 1402   class class class wbr 4130

This proof depends on 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 proof 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 3715  df-pr 3716  df-op 3718  df-br 4131

This theorem is used by:  enpr2d  7111  fiunsnnn  7185  exmidpw2en  7219  unsnfi  7226  2omapfi  7321  eninl  7438  eninr  7439  difinfinf  7442  exmidfodomrlemr  7555  exmidfodomrlemrALT  7556  dju1en  7570  djucomen  7573  djuassen  7574  xpdjuen  7575  gtndiv  9746  intqfrac2  10770  uzenom  10876  xrmaxiflemval  12034  ege2le3  12456  eirraplem  12562  bitsfzo  12740  pcprendvds  13091  pcpremul  13094  pcfaclem  13150  infpnlem2  13161  2strstr1g  13527  lmcn2  15433  dveflem  15879  tangtx  15992  ioocosf1o  16008  bposlem1  16233  bposlem2  16234  bposlem3  16235  lgsdirprm  16275  sbthom  17193  nconstwlpolemgt0  17236