Theorem eleqtrid 2320

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 2310	1 |- ( ph -> A e. C )

Syntax hints:   -> wi 4   = wceq 1398   e. wcel 2202

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 2213

This theorem depends on definitions:  df-bi 117  df-cleq 2224  df-clel 2227

This theorem is referenced by:  eleqtrrid  2321  opth1  4334  opth  4335  eqelsuc  4522  2omotaplemst  7537  txdis  15088  bj-nnelirr  16669