Theorem 3eltr4i 2844

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 2830	. 2 ⊢ 𝐴 ∈ 𝐷
5	1, 4	eqeltri 2827	1 ⊢ 𝐶 ∈ 𝐷

Syntax hints:   = wceq 1539   ∈ wcel 2104

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701

This theorem depends on definitions:  df-bi 206  df-an 395  df-ex 1780  df-cleq 2722  df-clel 2808

This theorem is referenced by:  oancom  9648  0r  11077  1sr  11078  m1r  11079  smndex1ibas  18817  recvs  24893  recvsOLD  24894  qcvs  24895  wlk2v2elem1  29675  konigsbergiedgw  29768  lmxrge0  33230  brsigarn  33480  ex-sategoelel12  34716  sinccvglem  34955  bj-minftyccb  36409  omcl3g  42386  fouriersw  45245