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Theorem syl6eleq 2177
Description: A membership and equality inference. (Contributed by NM, 4-Jan-2006.)
Hypotheses
Ref Expression
syl6eleq.1  |-  ( ph  ->  A  e.  B )
syl6eleq.2  |-  B  =  C
Assertion
Ref Expression
syl6eleq  |-  ( ph  ->  A  e.  C )

Proof of Theorem syl6eleq
StepHypRef Expression
1 syl6eleq.1 . 2  |-  ( ph  ->  A  e.  B )
2 syl6eleq.2 . . 3  |-  B  =  C
32a1i 9 . 2  |-  ( ph  ->  B  =  C )
41, 3eleqtrd 2163 1  |-  ( ph  ->  A  e.  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1287    e. wcel 1436
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1379  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-4 1443  ax-17 1462  ax-ial 1470  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-cleq 2078  df-clel 2081
This theorem is referenced by:  syl6eleqr  2178  prid2g  3530  freccllem  6121  finomni  6740  exmidomniim  6741  caucvgprprlem2  7213  gt0srpr  7238  eluzel2  8956  fseq1p1m1  9438  fznn0sub2  9467  nn0split  9475  nnsplit  9476  exple1  9910  ibcval5  10068  bcpasc  10071  zfz1isolemsplit  10140  iseqcoll  10144  clim2iser  10619  clim2iser2  10620  iiserex  10621  iisermulc2  10622  iserile  10624  iserige0  10625  climub  10626  climserile  10627  serif0  10633  isummolem3  10661  isummolem2a  10662  fisum  10667  sum0  10668  fsumcl2lem  10677  mod2eq1n2dvds  10761  ialgrp1  10910  phiprmpw  11080  crth  11082  phimullem  11083  nnsf  11340  nninfsellemqall  11352  nninfomnilem  11355
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