Theorem breqtrrdi 4076

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 2200	. 2 |- B = C
4	1, 3	breqtrdi 4075	1 |- ( ph -> A R C )

Syntax hints:   -> wi 4   = wceq 1364   class class class wbr 4034

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-un 3161  df-sn 3629  df-pr 3630  df-op 3632  df-br 4035

This theorem is referenced by:  enpr2d  6885  fiunsnnn  6951  exmidpw2en  6982  unsnfi  6989  eninl  7172  eninr  7173  difinfinf  7176  exmidfodomrlemr  7283  exmidfodomrlemrALT  7284  dju1en  7298  djucomen  7301  djuassen  7302  xpdjuen  7303  gtndiv  9440  intqfrac2  10430  uzenom  10536  xrmaxiflemval  11434  ege2le3  11855  eirraplem  11961  bitsfzo  12139  pcprendvds  12486  pcpremul  12489  pcfaclem  12545  infpnlem2  12556  2strstr1g  12826  lmcn2  14624  dveflem  15070  tangtx  15182  ioocosf1o  15198  lgsdirprm  15383  sbthom  15783  nconstwlpolemgt0  15821