Theorem 3eltr3d 2290

Description: Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.)

Hypotheses

Ref	Expression
3eltr3d.1	|- ( ph -> A e. B )
3eltr3d.2	|- ( ph -> A = C )
3eltr3d.3	|- ( ph -> B = D )

Assertion

Ref	Expression
3eltr3d	|- ( ph -> C e. D )

Proof of Theorem 3eltr3d

Step	Hyp	Ref	Expression
1		3eltr3d.2	. 2 |- ( ph -> A = C )
2		3eltr3d.1	. . 3 |- ( ph -> A e. B )
3		3eltr3d.3	. . 3 |- ( ph -> B = D )
4	2, 3	eleqtrd 2286	. 2 |- ( ph -> A e. D )
5	1, 4	eqeltrrd 2285	1 |- ( ph -> C e. D )

Syntax hints:   -> wi 4   = wceq 1373   e. wcel 2178

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 1471  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-4 1534  ax-17 1550  ax-ial 1558  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-cleq 2200  df-clel 2203

This theorem is referenced by:  reg3exmidlemwe  4645  nnaordi  6617  icoshftf1o  10148  lincmb01cmp  10160  fzosubel  10360  cnmpt2res  14884  dvcnp2cntop  15286