Theorem 3eltr4d 2318

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 2314	. 2 |- ( ph -> A e. D )
5	1, 4	eqeltrd 2311	1 |- ( ph -> C e. D )

Syntax hints:   -> wi 4   = wceq 1398   e. wcel 2205

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 2216

This theorem depends on definitions:  df-bi 117  df-cleq 2227  df-clel 2230

This theorem is referenced by:  ovmpodxf  6187  nnaordi  6754  iccf1o  10357  infssfzcldc  10618  ccatw2s1p1g  11358  nnmindc  12755  ennnfonelemrn  13254  ctiunctlemfo  13274  sgrppropd  13676  mndpropd  13701  issubmnd  13703  imasgrp  13864  mulgnndir  13904  subg0cl  13935  subginvcl  13936  subgcl  13937  rngcl  14183  rngpropd  14194  srgcl  14213  srgidcl  14219  ringidcl  14263  ringpropd  14281  dvdsrd  14339  dvrvald  14379  subrngmcl  14455  subrgmcl  14479  subrgunit  14485  lmodprop2d  14622  lidl0  14763  lidl1  14764  psraddcl  14961  wlkl1loop  16479  wlkres  16500  clwwlknonex2lem1  16558