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Theorem syl6eqbrr 4335
Description: A chained equality inference for a binary relation. (Contributed by NM, 4-Jan-2006.)
Hypotheses
Ref Expression
syl6eqbrr.1  |-  ( ph  ->  B  =  A )
syl6eqbrr.2  |-  B R C
Assertion
Ref Expression
syl6eqbrr  |-  ( ph  ->  A R C )

Proof of Theorem syl6eqbrr
StepHypRef Expression
1 syl6eqbrr.1 . . 3  |-  ( ph  ->  B  =  A )
21eqcomd 2448 . 2  |-  ( ph  ->  A  =  B )
3 syl6eqbrr.2 . 2  |-  B R C
42, 3syl6eqbr 4334 1  |-  ( ph  ->  A R C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1369   class class class wbr 4297
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-rab 2729  df-v 2979  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-sn 3883  df-pr 3885  df-op 3889  df-br 4298
This theorem is referenced by:  grur1  8992  t1conperf  19045  basellem9  22431  sqff1o  22525  ballotlemic  26894  ballotlem1c  26895
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