Theorem 3eltr4i 2875

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 2861	. 2 ⊢ 𝐴 ∈ 𝐷
5	1, 4	eqeltri 2858	1 ⊢ 𝐶 ∈ 𝐷

This proof depends on syntax axioms:   = wceq 1570   ∈ wcel 2145

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-cleq 2754  df-clel 2837

This theorem is used by:  oancom  9633  0r  11092  1sr  11093  m1r  11094  smndex1ibas  19010  recvs  25375  qcvs  25376  wlk2v2elem1  30621  konigsbergiedgw  30714  lmxrge0  34449  brsigarn  34682  ex-sategoelel12  35993  sinccvglem  36238  bj-minftyccb  37964  resuppsinopn  43225  omcl3g  44162  fouriersw  47046