Theorem eleqtrrid 2286

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 2202	. 2 |- ( ph -> B = C )
4	1, 3	eleqtrid 2285	1 |- ( ph -> A e. C )

Syntax hints:   -> wi 4   = wceq 1364   e. wcel 2167

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 1461  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-4 1524  ax-17 1540  ax-ial 1548  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-cleq 2189  df-clel 2192

This theorem is referenced by:  rabsnt  3698  exmid1stab  4242  0elnn  4656  canth  5878  tfrexlem  6401  rdgtfr  6441  rdgruledefgg  6442  exmidonfinlem  7272  hashinfom  10887  ennnfonelemhom  12657  fnpr2ob  13042