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Theorem syl5breq 4482
Description: A chained equality inference for a binary relation. (Contributed by NM, 11-Oct-1999.)
Hypotheses
Ref Expression
syl5breq.1  |-  A R B
syl5breq.2  |-  ( ph  ->  B  =  C )
Assertion
Ref Expression
syl5breq  |-  ( ph  ->  A R C )

Proof of Theorem syl5breq
StepHypRef Expression
1 syl5breq.1 . . 3  |-  A R B
21a1i 11 . 2  |-  ( ph  ->  A R B )
3 syl5breq.2 . 2  |-  ( ph  ->  B  =  C )
42, 3breqtrd 4471 1  |-  ( ph  ->  A R C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379   class class class wbr 4447
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448
This theorem is referenced by:  syl5breqr  4483  phplem3  7695  xlemul1a  11476  phicl2  14150  sinq12ge0  22631  siilem1  25439  nmbdfnlbi  26641  nmcfnlbi  26644  unierri  26696  leoprf2  26719  leoprf  26720  ballotlemic  28082  ballotlem1c  28083
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