Theorem 3eltr4d 2280

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 2276	. 2 |- ( ph -> A e. D )
5	1, 4	eqeltrd 2273	1 |- ( ph -> C e. D )

Syntax hints:   -> wi 4   = wceq 1364   e. wcel 2167

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 1461  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-4 1524  ax-17 1540  ax-ial 1548  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-cleq 2189  df-clel 2192

This theorem is referenced by:  ovmpodxf  6052  nnaordi  6575  iccf1o  10096  nnmindc  12226  ennnfonelemrn  12661  ctiunctlemfo  12681  sgrppropd  13115  mndpropd  13142  issubmnd  13144  imasgrp  13317  mulgnndir  13357  subg0cl  13388  subginvcl  13389  subgcl  13390  rngcl  13576  rngpropd  13587  srgcl  13602  srgidcl  13608  ringidcl  13652  ringpropd  13670  dvdsrd  13726  dvrvald  13766  subrngmcl  13841  subrgmcl  13865  subrgunit  13871  lmodprop2d  13980  lidl0  14121  lidl1  14122  psraddcl  14308