Theorem eleqtrrid 2319

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 2235	. 2 |- ( ph -> B = C )
4	1, 3	eleqtrid 2318	1 |- ( ph -> A e. C )

Syntax hints:   -> wi 4   = wceq 1395   e. wcel 2200

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 1493  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-4 1556  ax-17 1572  ax-ial 1580  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-cleq 2222  df-clel 2225

This theorem is referenced by:  rabsnt  3741  exmid1stab  4292  0elnn  4711  canth  5952  tfrexlem  6480  rdgtfr  6520  rdgruledefgg  6521  exmidonfinlem  7371  hashinfom  11000  swrds1  11200  ennnfonelemhom  12986  fnpr2ob  13373  upgrex  15903  wlkl1loop  16069