Theorem breqtrrdi 4087

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 2209	. 2 |- B = C
4	1, 3	breqtrdi 4086	1 |- ( ph -> A R C )

Syntax hints:   -> wi 4   = wceq 1373   class class class wbr 4045

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-un 3170  df-sn 3639  df-pr 3640  df-op 3642  df-br 4046

This theorem is referenced by:  enpr2d  6913  fiunsnnn  6980  exmidpw2en  7011  unsnfi  7018  eninl  7201  eninr  7202  difinfinf  7205  exmidfodomrlemr  7312  exmidfodomrlemrALT  7313  dju1en  7327  djucomen  7330  djuassen  7331  xpdjuen  7332  gtndiv  9470  intqfrac2  10466  uzenom  10572  xrmaxiflemval  11594  ege2le3  12015  eirraplem  12121  bitsfzo  12299  pcprendvds  12646  pcpremul  12649  pcfaclem  12705  infpnlem2  12716  2strstr1g  12987  lmcn2  14785  dveflem  15231  tangtx  15343  ioocosf1o  15359  lgsdirprm  15544  sbthom  16002  nconstwlpolemgt0  16040