Theorem eleqtrrid 2324

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 2240	. 2 |- ( ph -> B = C )
4	1, 3	eleqtrid 2323	1 |- ( ph -> A e. C )

Syntax hints:   -> wi 4   = wceq 1398   e. wcel 2205

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 1496  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-4 1559  ax-17 1575  ax-ial 1583  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-cleq 2227  df-clel 2230

This theorem is referenced by:  rabsnt  3768  exmid1stab  4323  0elnn  4743  canth  6003  tfrexlem  6567  rdgtfr  6607  rdgruledefgg  6608  exmidonfinlem  7498  hashinfom  11145  swrds1  11364  ennnfonelemhom  13183  fnpr2ob  13570  upgrex  16115  upgr1een  16136  wlkl1loop  16370