Theorem 3eltr4i 2845

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 2831	. 2 ⊢ 𝐴 ∈ 𝐷
5	1, 4	eqeltri 2828	1 ⊢ 𝐶 ∈ 𝐷

Syntax hints:   = wceq 1541   ∈ wcel 2106

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1782  df-cleq 2723  df-clel 2809

This theorem is referenced by:  oancom  9596  0r  11025  1sr  11026  m1r  11027  smndex1ibas  18724  recvs  24546  recvsOLD  24547  qcvs  24548  wlk2v2elem1  29162  konigsbergiedgw  29255  lmxrge0  32622  brsigarn  32872  ex-sategoelel12  34108  sinccvglem  34347  bj-minftyccb  35769  omcl3g  41727  fouriersw  44592