Theorem eleqtrid 2282

Description: B membership and equality inference. (Contributed by NM, 4-Jan-2006.)

Hypotheses

Ref	Expression
eleqtrid.1	|- A e. B
eleqtrid.2	|- ( ph -> B = C )

Assertion

Ref	Expression
eleqtrid	|- ( ph -> A e. C )

Proof of Theorem eleqtrid

Step	Hyp	Ref	Expression
1		eleqtrid.1	. . 3 |- A e. B
2	1	a1i 9	. 2 |- ( ph -> A e. B )
3		eleqtrid.2	. 2 |- ( ph -> B = C )
4	2, 3	eleqtrd 2272	1 |- ( ph -> A e. C )

Syntax hints:   -> wi 4   = wceq 1364   e. wcel 2164

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 2175

This theorem depends on definitions:  df-bi 117  df-cleq 2186  df-clel 2189

This theorem is referenced by:  eleqtrrid  2283  opth1  4265  opth  4266  eqelsuc  4450  2omotaplemst  7318  txdis  14445  bj-nnelirr  15445