Theorem eleqtrrid 2279

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 2195	. 2 |- ( ph -> B = C )
4	1, 3	eleqtrid 2278	1 |- ( ph -> A e. C )

Syntax hints:   -> wi 4   = wceq 1364   e. wcel 2160

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 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-4 1521  ax-17 1537  ax-ial 1545  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-cleq 2182  df-clel 2185

This theorem is referenced by:  rabsnt  3682  exmid1stab  4226  0elnn  4636  canth  5849  tfrexlem  6358  rdgtfr  6398  rdgruledefgg  6399  exmidonfinlem  7221  hashinfom  10789  ennnfonelemhom  12465  fnpr2ob  12813