Theorem 3eltr4i 2841

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 2827	. 2 ⊢ 𝐴 ∈ 𝐷
5	1, 4	eqeltri 2824	1 ⊢ 𝐶 ∈ 𝐷

Syntax hints:   = wceq 1540   ∈ wcel 2109

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 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-cleq 2721  df-clel 2803

This theorem is referenced by:  oancom  9547  0r  10974  1sr  10975  m1r  10976  smndex1ibas  18774  recvs  25044  qcvs  25045  wlk2v2elem1  30099  konigsbergiedgw  30192  lmxrge0  33925  brsigarn  34157  ex-sategoelel12  35410  sinccvglem  35655  bj-minftyccb  37209  resuppsinopn  42346  omcl3g  43317  fouriersw  46222