Theorem 3eltr4i 2926

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 2912	. 2 ⊢ 𝐴 ∈ 𝐷
5	1, 4	eqeltri 2909	1 ⊢ 𝐶 ∈ 𝐷

Syntax hints:   = wceq 1533   ∈ wcel 2110

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1777  df-cleq 2814  df-clel 2893

This theorem is referenced by:  oancom  9108  0r  10496  1sr  10497  m1r  10498  smndex1ibas  18059  recvs  23744  qcvs  23745  wlk2v2elem1  27928  konigsbergiedgw  28021  lmxrge0  31190  brsigarn  31438  ex-sategoelel12  32669  sinccvglem  32910  bj-minftyccb  34501  fouriersw  42510