Theorem 3eltr4i 2865

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 2851	. 2 ⊢ 𝐴 ∈ 𝐷
5	1, 4	eqeltri 2848	1 ⊢ 𝐶 ∈ 𝐷

Syntax hints:   = wceq 1550   ∈ wcel 2132

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1805  ax-4 1819  ax-5 1920  ax-6 1977  ax-7 2018  ax-8 2134  ax-9 2142  ax-ext 2724

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1790  df-cleq 2744  df-clel 2827

This theorem is referenced by:  oancom  9592  0r  11024  1sr  11025  m1r  11026  smndex1ibas  18906  recvs  25177  qcvs  25178  wlk2v2elem1  30292  konigsbergiedgw  30385  lmxrge0  34193  brsigarn  34425  ex-sategoelel12  35715  sinccvglem  35960  bj-minftyccb  37655  resuppsinopn  42910  omcl3g  43849  fouriersw  46743