MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  syl5eleq Structured version   Unicode version

Theorem syl5eleq 2551
Description: A 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 2547 1  |-  ( ph  ->  A  e.  C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1395    e. wcel 1819
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-ext 2435
This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1614  df-cleq 2449  df-clel 2452
This theorem is referenced by:  syl5eleqr  2552  opth1  4729  opth  4730  eqelsuc  4968  tfrlem11  7075  oalimcl  7227  omlimcl  7245  frgp0  16904  txdis  20258  ordtconlem1  28059  rankeq1o  29990
  Copyright terms: Public domain W3C validator