Theorem 3eltr4i 2850

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 2836	. 2 ⊢ 𝐴 ∈ 𝐷
5	1, 4	eqeltri 2833	1 ⊢ 𝐶 ∈ 𝐷

Syntax hints:   = wceq 1542   ∈ wcel 2114

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1782  df-cleq 2729  df-clel 2812

This theorem is referenced by:  oancom  9564  0r  10995  1sr  10996  m1r  10997  smndex1ibas  18829  recvs  25106  qcvs  25107  wlk2v2elem1  30213  konigsbergiedgw  30306  lmxrge0  34090  brsigarn  34322  ex-sategoelel12  35602  sinccvglem  35847  bj-minftyccb  37401  resuppsinopn  42654  omcl3g  43612  fouriersw  46511