Theorem 3eltr4d 2162

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 2158	. 2 |- ( ph -> A e. D )
5	1, 4	eqeltrd 2155	1 |- ( ph -> C e. D )

Syntax hints:   -> wi 4   = wceq 1284   e. wcel 1433

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-4 1440  ax-17 1459  ax-ial 1467  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-cleq 2074  df-clel 2077

This theorem is referenced by:  ovmpt2dxf  5646  nnaordi  6104  iccf1o  9026