Theorem breqtrrdi 4173

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 4172	1 |- ( ph -> A R C )

This proof depends on syntax axioms:   -> wi 4   = wceq 1402   class class class wbr 4131

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 4132

This theorem is used by:  enpr2d  7117  fiunsnnn  7191  exmidpw2en  7225  unsnfi  7232  2omapfi  7327  eninl  7444  eninr  7445  difinfinf  7448  exmidfodomrlemr  7561  exmidfodomrlemrALT  7562  dju1en  7576  djucomen  7579  djuassen  7580  xpdjuen  7581  gtndiv  9752  intqfrac2  10777  uzenom  10883  xrmaxiflemval  12041  ege2le3  12463  eirraplem  12569  bitsfzo  12747  pcprendvds  13098  pcpremul  13101  pcfaclem  13157  infpnlem2  13168  2strstr1g  13535  lmcn2  15539  dveflem  15985  tangtx  16098  ioocosf1o  16115  bposlem1  16344  bposlem2  16345  bposlem3  16346  lgsdirprm  16391  sbthom  17310  nconstwlpolemgt0  17354