Theorem 3eltr4i 2876

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 2862	. 2 ⊢ 𝐴 ∈ 𝐷
5	1, 4	eqeltri 2859	1 ⊢ 𝐶 ∈ 𝐷

Syntax hints:   = wceq 1561   ∈ wcel 2143

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 209  df-an 400  df-ex 1801  df-cleq 2755  df-clel 2838

This theorem is referenced by:  oancom  9604  0r  11049  1sr  11050  m1r  11051  smndex1ibas  18944  recvs  25215  qcvs  25216  wlk2v2elem1  30364  konigsbergiedgw  30457  lmxrge0  34251  brsigarn  34483  ex-sategoelel12  35782  sinccvglem  36027  bj-minftyccb  37722  resuppsinopn  42977  omcl3g  43916  fouriersw  46796