Theorem 3eltr4i 2872

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 2858	. 2 ⊢ 𝐴 ∈ 𝐷
5	1, 4	eqeltri 2855	1 ⊢ 𝐶 ∈ 𝐷

Syntax hints:   = wceq 1601   ∈ wcel 2107

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-ext 2754

This theorem depends on definitions:  df-bi 199  df-an 387  df-ex 1824  df-cleq 2770  df-clel 2774

This theorem is referenced by:  oancom  8845  0r  10237  1sr  10238  m1r  10239  recvs  23353  qcvs  23354  wlk2v2elem1  27558  konigsbergiedgw  27654  lmxrge0  30596  brsigarn  30845  sinccvglem  32163  bj-minftyccb  33702  fouriersw  41375