Theorem eleqtrid 2294

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 2284	1 |- ( ph -> A e. C )

Syntax hints:   -> wi 4   = wceq 1373   e. wcel 2176

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 1470  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-4 1533  ax-17 1549  ax-ial 1557  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-cleq 2198  df-clel 2201

This theorem is referenced by:  eleqtrrid  2295  opth1  4281  opth  4282  eqelsuc  4467  2omotaplemst  7372  txdis  14782  bj-nnelirr  15926