Theorem 3eltr4i 2926

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 2912	. 2 ⊢ 𝐴 ∈ 𝐷
5	1, 4	eqeltri 2909	1 ⊢ 𝐶 ∈ 𝐷

Syntax hints:   = wceq 1537   ∈ wcel 2114

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1781  df-cleq 2814  df-clel 2893

This theorem is referenced by:  oancom  9114  0r  10502  1sr  10503  m1r  10504  smndex1ibas  18065  recvs  23750  qcvs  23751  wlk2v2elem1  27934  konigsbergiedgw  28027  lmxrge0  31195  brsigarn  31443  ex-sategoelel12  32674  sinccvglem  32915  bj-minftyccb  34510  fouriersw  42536