Theorem syl6eleq 2171

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 2157	1 |- ( ph -> A e. C )

Syntax hints:   -> wi 4   = wceq 1284   e. wcel 1433

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 1376  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-4 1440  ax-17 1459  ax-ial 1467  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-cleq 2074  df-clel 2077

This theorem is referenced by:  syl6eleqr  2172  prid2g  3497  caucvgprprlem2  6900  gt0srpr  6925  eluzel2  8624  fseq1p1m1  9111  fznn0sub2  9139  nn0split  9147  exple1  9532  ibcval5  9690  bcpasc  9693  clim2iser  10175  clim2iser2  10176  iiserex  10177  iisermulc2  10178  iserile  10180  iserige0  10181  climub  10182  climserile  10183  serif0  10189  mod2eq1n2dvds  10279  ialgrp1  10428