Theorem 3eltr4d 2316

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 2312	. 2 |- ( ph -> A e. D )
5	1, 4	eqeltrd 2309	1 |- ( ph -> C e. D )

Syntax hints:   -> wi 4   = wceq 1398   e. wcel 2203

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 2214

This theorem depends on definitions:  df-bi 117  df-cleq 2225  df-clel 2228

This theorem is referenced by:  ovmpodxf  6179  nnaordi  6741  iccf1o  10338  ccatw2s1p1g  11333  nnmindc  12730  ennnfonelemrn  13170  ctiunctlemfo  13190  sgrppropd  13626  mndpropd  13653  issubmnd  13655  imasgrp  13828  mulgnndir  13868  subg0cl  13899  subginvcl  13900  subgcl  13901  rngcl  14088  rngpropd  14099  srgcl  14114  srgidcl  14120  ringidcl  14164  ringpropd  14182  dvdsrd  14239  dvrvald  14279  subrngmcl  14354  subrgmcl  14378  subrgunit  14384  lmodprop2d  14496  lidl0  14637  lidl1  14638  psraddcl  14835  wlkl1loop  16353  wlkres  16374  clwwlknonex2lem1  16432