Theorem syl6eleq 2172

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

Step	Hyp	Ref	Expression
1		syl6eleq.1	. 2 |- ( ph -> A e. B )
2		syl6eleq.2	. . 3 |- B = C
3	2	a1i 9	. 2 |- ( ph -> B = C )
4	1, 3	eleqtrd 2158	1 |- ( ph -> A e. C )

Syntax hints:   -> wi 4   = wceq 1285   e. wcel 1434

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 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-4 1441  ax-17 1460  ax-ial 1468  ax-ext 2064

This theorem depends on definitions:  df-bi 115  df-cleq 2075  df-clel 2078

This theorem is referenced by:  syl6eleqr  2173  prid2g  3505  freccllem  6051  caucvgprprlem2  6962  gt0srpr  6987  eluzel2  8705  fseq1p1m1  9187  fznn0sub2  9216  nn0split  9224  exple1  9629  ibcval5  9787  bcpasc  9790  clim2iser  10313  clim2iser2  10314  iiserex  10315  iisermulc2  10316  iserile  10318  iserige0  10319  climub  10320  climserile  10321  serif0  10327  mod2eq1n2dvds  10423  ialgrp1  10572