Theorem 3eltr4i 2857

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 2843	. 2 ⊢ 𝐴 ∈ 𝐷
5	1, 4	eqeltri 2840	1 ⊢ 𝐶 ∈ 𝐷

Syntax hints:   = wceq 1537   ∈ wcel 2108

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1778  df-cleq 2732  df-clel 2819

This theorem is referenced by:  oancom  9720  0r  11149  1sr  11150  m1r  11151  smndex1ibas  18935  recvs  25198  recvsOLD  25199  qcvs  25200  wlk2v2elem1  30187  konigsbergiedgw  30280  lmxrge0  33898  brsigarn  34148  ex-sategoelel12  35395  sinccvglem  35640  bj-minftyccb  37191  omcl3g  43296  fouriersw  46152