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Theorem syl5breq 4249
Description: B 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 4238 1  |-  ( ph  ->  A R C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653   class class class wbr 4214
This theorem is referenced by:  syl5breqr  4250  phplem3  7290  xlemul1a  10869  phicl2  13159  sinq12ge0  20418  siilem1  22354  nmbdfnlbi  23554  nmcfnlbi  23557  unierri  23609  leoprf2  23632  leoprf  23633  ballotlemic  24766  ballotlem1c  24767
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-br 4215
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