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Theorem syl5breq 4429
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 4418 1  |-  ( ph  ->  A R C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1405   class class class wbr 4394
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-rab 2762  df-v 3060  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-br 4395
This theorem is referenced by:  syl5breqr  4430  phplem3  7656  xlemul1a  11451  phicl2  14399  sinq12ge0  23085  siilem1  26060  nmbdfnlbi  27261  nmcfnlbi  27264  unierri  27316  leoprf2  27339  leoprf  27340  ballotlemic  28831  ballotlem1c  28832  sumnnodd  36986
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