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Theorem syl5eleq 2382
Description: B membership and equality inference. (Contributed by NM, 4-Jan-2006.)
Hypotheses
Ref Expression
syl5eleq.1  |-  A  e.  B
syl5eleq.2  |-  ( ph  ->  B  =  C )
Assertion
Ref Expression
syl5eleq  |-  ( ph  ->  A  e.  C )

Proof of Theorem syl5eleq
StepHypRef Expression
1 syl5eleq.1 . . 3  |-  A  e.  B
21a1i 10 . 2  |-  ( ph  ->  A  e.  B )
3 syl5eleq.2 . 2  |-  ( ph  ->  B  =  C )
42, 3eleqtrd 2372 1  |-  ( ph  ->  A  e.  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696
This theorem is referenced by:  syl5eleqr  2383  opth1  4260  opth  4261  eqelsuc  4489  tfrlem11  6420  oalimcl  6574  omlimcl  6592  frgp0  15085  txdis  17342  rankeq1o  24873
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-11 1727  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1532  df-cleq 2289  df-clel 2292
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