MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  syl5breq Structured version   Unicode version

Theorem syl5breq 4434
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 4423 1  |-  ( ph  ->  A R C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370   class class class wbr 4399
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-rab 2807  df-v 3078  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-nul 3745  df-if 3899  df-sn 3985  df-pr 3987  df-op 3991  df-br 4400
This theorem is referenced by:  syl5breqr  4435  phplem3  7601  xlemul1a  11361  phicl2  13960  sinq12ge0  22102  siilem1  24402  nmbdfnlbi  25604  nmcfnlbi  25607  unierri  25659  leoprf2  25682  leoprf  25683  ballotlemic  27032  ballotlem1c  27033
  Copyright terms: Public domain W3C validator