Theorem 3eltr4i 2854

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 2840	. 2 ⊢ 𝐴 ∈ 𝐷
5	1, 4	eqeltri 2837	1 ⊢ 𝐶 ∈ 𝐷

Syntax hints:   = wceq 1540   ∈ wcel 2108

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 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-cleq 2729  df-clel 2816

This theorem is referenced by:  oancom  9691  0r  11120  1sr  11121  m1r  11122  smndex1ibas  18913  recvs  25179  recvsOLD  25180  qcvs  25181  wlk2v2elem1  30174  konigsbergiedgw  30267  lmxrge0  33951  brsigarn  34185  ex-sategoelel12  35432  sinccvglem  35677  bj-minftyccb  37226  resuppsinopn  42393  omcl3g  43347  fouriersw  46246