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 1541   ∈ wcel 2111

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1781  df-cleq 2723  df-clel 2806

This theorem is referenced by:  oancom  9547  0r  10977  1sr  10978  m1r  10979  smndex1ibas  18814  recvs  25079  qcvs  25080  wlk2v2elem1  30142  konigsbergiedgw  30235  lmxrge0  33972  brsigarn  34204  ex-sategoelel12  35478  sinccvglem  35723  bj-minftyccb  37276  resuppsinopn  42462  omcl3g  43432  fouriersw  46334