Theorem eleqtrrid 2328

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 2244	. 2 |- ( ph -> B = C )
4	1, 3	eleqtrid 2327	1 |- ( ph -> A e. C )

Syntax hints:   -> wi 4   = wceq 1402   e. wcel 2209

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 1500  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-4 1563  ax-17 1579  ax-ial 1587  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-cleq 2231  df-clel 2234

This theorem is referenced by:  rabsnt  3782  exmid1stab  4340  0elnn  4761  canth  6026  tfrexlem  6595  rdgtfr  6635  rdgruledefgg  6636  exmidonfinlem  7535  hashinfom  11195  swrds1  11418  ennnfonelemhom  13284  fnpr2ob  13638  upgrex  16258  upgr1een  16279  wlkl1loop  16513