Theorem eleqtrrid 2297

Description: B membership and equality inference. (Contributed by NM, 4-Jan-2006.)

Hypotheses

Ref	Expression
eleqtrrid.1	|- A e. B
eleqtrrid.2	|- ( ph -> C = B )

Assertion

Ref	Expression
eleqtrrid	|- ( ph -> A e. C )

Proof of Theorem eleqtrrid

Step	Hyp	Ref	Expression
1		eleqtrrid.1	. 2 |- A e. B
2		eleqtrrid.2	. . 3 |- ( ph -> C = B )
3	2	eqcomd 2213	. 2 |- ( ph -> B = C )
4	1, 3	eleqtrid 2296	1 |- ( ph -> A e. C )

Syntax hints:   -> wi 4   = wceq 1373   e. wcel 2178

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1471  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-4 1534  ax-17 1550  ax-ial 1558  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-cleq 2200  df-clel 2203

This theorem is referenced by:  rabsnt  3718  exmid1stab  4268  0elnn  4685  canth  5920  tfrexlem  6443  rdgtfr  6483  rdgruledefgg  6484  exmidonfinlem  7332  hashinfom  10960  swrds1  11159  ennnfonelemhom  12901  fnpr2ob  13287  upgrex  15814