Theorem eleqtrrd 2168

Description: Deduction that substitutes equal classes into membership. (Contributed by NM, 14-Dec-2004.)

Hypotheses

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

Assertion

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

Proof of Theorem eleqtrrd

Step	Hyp	Ref	Expression
1		eleqtrrd.1	. 2 |- ( ph -> A e. B )
2		eleqtrrd.2	. . 3 |- ( ph -> C = B )
3	2	eqcomd 2094	. 2 |- ( ph -> B = C )
4	1, 3	eleqtrd 2167	1 |- ( ph -> A e. C )

Syntax hints:   -> wi 4   = wceq 1290   e. wcel 1439

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1382  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-4 1446  ax-17 1465  ax-ial 1473  ax-ext 2071

This theorem depends on definitions:  df-bi 116  df-cleq 2082  df-clel 2085

This theorem is referenced by:  3eltr4d  2172  tfrexlem  6113  nnsucuniel  6270  erref  6326  en1uniel  6575  fin0  6655  fin0or  6656  prarloclemarch2  7039  fzopth  9536  fzoss2  9644  fzosubel3  9668  elfzomin  9678  elfzonlteqm1  9682  fzoend  9694  fzofzp1  9699  fzofzp1b  9700  peano2fzor  9704  zmodfzo  9815  frecuzrdg0  9881  frecuzrdgsuc  9882  frecuzrdgdomlem  9885  frecuzrdg0t  9890  frecuzrdgsuctlem  9891  seq3f1olemqsum  9990  bcn2  10233  isummolem2a  10832  fisumss  10845  fsumm1  10871  fisumcom2  10893