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Theorem syl5breq 4058
Description: B 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 10 . 2  |-  ( ph  ->  A R B )
3 syl5breq.2 . 2  |-  ( ph  ->  B  =  C )
42, 3breqtrd 4047 1  |-  ( ph  ->  A R C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623   class class class wbr 4023
This theorem is referenced by:  syl5breqr  4059  phplem3  7042  xlemul1a  10608  phicl2  12836  sinq12ge0  19876  siilem1  21429  nmbdfnlbi  22629  nmcfnlbi  22632  unierri  22684  leoprf2  22707  leoprf  22708  stoweidlem55  27804
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024
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