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 1508   ∈ wcel 2051

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-ext 2743

This theorem depends on definitions:  df-bi 199  df-an 388  df-ex 1744  df-cleq 2764  df-clel 2839

This theorem is referenced by:  oancom  8906  0r  10298  1sr  10299  m1r  10300  recvs  23468  qcvs  23469  wlk2v2elem1  27699  konigsbergiedgw  27795  lmxrge0  30871  brsigarn  31120  sinccvglem  32472  bj-minftyccb  34013  fouriersw  41981