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Theorem syl5eleq 2521
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 11 . 2  |-  ( ph  ->  A  e.  B )
3 syl5eleq.2 . 2  |-  ( ph  ->  B  =  C )
42, 3eleqtrd 2511 1  |-  ( ph  ->  A  e.  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725
This theorem is referenced by:  syl5eleqr  2522  opth1  4426  opth  4427  eqelsuc  4654  tfrlem11  6641  oalimcl  6795  omlimcl  6813  frgp0  15384  txdis  17656  rankeq1o  26104
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-11 1761  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1551  df-cleq 2428  df-clel 2431
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