Theorem 3eltr4i 2839

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

Hypotheses

Ref	Expression
3eltr4i.1	⊢ 𝐴 ∈ 𝐵
3eltr4i.2	⊢ 𝐶 = 𝐴
3eltr4i.3	⊢ 𝐷 = 𝐵

Assertion

Ref	Expression
3eltr4i	⊢ 𝐶 ∈ 𝐷

Proof of Theorem 3eltr4i

Step	Hyp	Ref	Expression
1		3eltr4i.2	. 2 ⊢ 𝐶 = 𝐴
2		3eltr4i.1	. . 3 ⊢ 𝐴 ∈ 𝐵
3		3eltr4i.3	. . 3 ⊢ 𝐷 = 𝐵
4	2, 3	eleqtrri 2825	. 2 ⊢ 𝐴 ∈ 𝐷
5	1, 4	eqeltri 2822	1 ⊢ 𝐶 ∈ 𝐷

Syntax hints:   = wceq 1534   ∈ wcel 2099

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697

This theorem depends on definitions:  df-bi 206  df-an 395  df-ex 1775  df-cleq 2718  df-clel 2803

This theorem is referenced by:  oancom  9694  0r  11123  1sr  11124  m1r  11125  smndex1ibas  18890  recvs  25164  recvsOLD  25165  qcvs  25166  wlk2v2elem1  30088  konigsbergiedgw  30181  lmxrge0  33767  brsigarn  34017  ex-sategoelel12  35255  sinccvglem  35500  bj-minftyccb  36932  omcl3g  43000  fouriersw  45852