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Theorem syl6eqbrr 4398
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 2428 . 2  |-  ( ph  ->  A  =  B )
3 syl6eqbrr.2 . 2  |-  B R C
42, 3syl6eqbr 4397 1  |-  ( ph  ->  A R C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1437   class class class wbr 4359
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2058  ax-ext 2402
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2552  df-rab 2717  df-v 3018  df-dif 3375  df-un 3377  df-in 3379  df-ss 3386  df-nul 3698  df-if 3848  df-sn 3935  df-pr 3937  df-op 3941  df-br 4360
This theorem is referenced by:  grur1  9189  t1conperf  20386  basellem9  23950  sqff1o  24044  ballotlemic  29284  ballotlem1c  29285  ballotlemicOLD  29322  ballotlem1cOLD  29323
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