Theorem breqtrrdi 4150

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 2236	. 2 |- B = C
4	1, 3	breqtrdi 4149	1 |- ( ph -> A R C )

Syntax hints:   -> wi 4   = wceq 1398   class class class wbr 4108

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2814  df-un 3214  df-sn 3694  df-pr 3695  df-op 3697  df-br 4109

This theorem is referenced by:  enpr2d  7063  fiunsnnn  7137  exmidpw2en  7171  unsnfi  7178  2omapfi  7270  eninl  7387  eninr  7388  difinfinf  7391  exmidfodomrlemr  7504  exmidfodomrlemrALT  7505  dju1en  7519  djucomen  7522  djuassen  7523  xpdjuen  7524  gtndiv  9669  intqfrac2  10677  uzenom  10783  xrmaxiflemval  11928  ege2le3  12350  eirraplem  12456  bitsfzo  12634  pcprendvds  12981  pcpremul  12984  pcfaclem  13040  infpnlem2  13051  2strstr1g  13324  lmcn2  15132  dveflem  15578  tangtx  15690  ioocosf1o  15706  lgsdirprm  15894  sbthom  16793  nconstwlpolemgt0  16836