Theorem 3eltr4d 2313

Description: Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.)

Hypotheses

Ref	Expression
3eltr4d.1	|- ( ph -> A e. B )
3eltr4d.2	|- ( ph -> C = A )
3eltr4d.3	|- ( ph -> D = B )

Assertion

Ref	Expression
3eltr4d	|- ( ph -> C e. D )

Proof of Theorem 3eltr4d

Step	Hyp	Ref	Expression
1		3eltr4d.2	. 2 |- ( ph -> C = A )
2		3eltr4d.1	. . 3 |- ( ph -> A e. B )
3		3eltr4d.3	. . 3 |- ( ph -> D = B )
4	2, 3	eleqtrrd 2309	. 2 |- ( ph -> A e. D )
5	1, 4	eqeltrd 2306	1 |- ( ph -> C e. D )

Syntax hints:   -> wi 4   = wceq 1395   e. wcel 2200

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 1493  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-4 1556  ax-17 1572  ax-ial 1580  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-cleq 2222  df-clel 2225

This theorem is referenced by:  ovmpodxf  6136  nnaordi  6662  iccf1o  10208  nnmindc  12563  ennnfonelemrn  12998  ctiunctlemfo  13018  sgrppropd  13454  mndpropd  13481  issubmnd  13483  imasgrp  13656  mulgnndir  13696  subg0cl  13727  subginvcl  13728  subgcl  13729  rngcl  13915  rngpropd  13926  srgcl  13941  srgidcl  13947  ringidcl  13991  ringpropd  14009  dvdsrd  14066  dvrvald  14106  subrngmcl  14181  subrgmcl  14205  subrgunit  14211  lmodprop2d  14320  lidl0  14461  lidl1  14462  psraddcl  14652  wlkl1loop  16079  wlkres  16098