Theorem 3eltr4d 2261

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 2257	. 2 |- ( ph -> A e. D )
5	1, 4	eqeltrd 2254	1 |- ( ph -> C e. D )

Syntax hints:   -> wi 4   = wceq 1353   e. wcel 2148

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 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-4 1510  ax-17 1526  ax-ial 1534  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-cleq 2170  df-clel 2173

This theorem is referenced by:  ovmpodxf  6003  nnaordi  6512  iccf1o  10007  nnmindc  12038  ennnfonelemrn  12423  ctiunctlemfo  12443  mndpropd  12847  issubmnd  12849  mulgnndir  13018  subg0cl  13048  subginvcl  13049  subgcl  13050  srgcl  13159  srgidcl  13165  ringidcl  13209  ringpropd  13223  dvdsrd  13269  dvrvald  13309  subrgmcl  13360  subrgunit  13366  lmodprop2d  13444  lidl0  13573  lidl1  13574