Theorem 3eltr4i 2874

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 2860	. 2 ⊢ 𝐴 ∈ 𝐷
5	1, 4	eqeltri 2857	1 ⊢ 𝐶 ∈ 𝐷

Syntax hints:   = wceq 1568   ∈ wcel 2141

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1808  df-cleq 2753  df-clel 2836

This theorem is referenced by:  oancom  9619  0r  11064  1sr  11065  m1r  11066  smndex1ibas  18958  recvs  25284  qcvs  25285  wlk2v2elem1  30472  konigsbergiedgw  30565  lmxrge0  34308  brsigarn  34540  ex-sategoelel12  35873  sinccvglem  36118  bj-minftyccb  37813  resuppsinopn  43070  omcl3g  44009  fouriersw  46893