Theorem 3eltr4i 2875

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 2861	. 2 ⊢ 𝐴 ∈ 𝐷
5	1, 4	eqeltri 2858	1 ⊢ 𝐶 ∈ 𝐷

This proof depends on syntax axioms:   = wceq 1569   ∈ wcel 2142

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-ex 1809  df-cleq 2754  df-clel 2837

This theorem is used by:  oancom  9618  0r  11071  1sr  11072  m1r  11073  smndex1ibas  18965  recvs  25316  qcvs  25317  wlk2v2elem1  30517  konigsbergiedgw  30610  lmxrge0  34351  brsigarn  34583  ex-sategoelel12  35927  sinccvglem  36172  bj-minftyccb  37897  resuppsinopn  43152  omcl3g  44089  fouriersw  46973