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Theorem syl6eqbrr 4494
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 2465 . 2  |-  ( ph  ->  A  =  B )
3 syl6eqbrr.2 . 2  |-  B R C
42, 3syl6eqbr 4493 1  |-  ( ph  ->  A R C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1395   class class class wbr 4456
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-br 4457
This theorem is referenced by:  grur1  9215  t1conperf  20062  basellem9  23487  sqff1o  23581  ballotlemic  28620  ballotlem1c  28621
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