Theorem 3eltr4i 2847

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 2833	. 2 ⊢ 𝐴 ∈ 𝐷
5	1, 4	eqeltri 2830	1 ⊢ 𝐶 ∈ 𝐷

Syntax hints:   = wceq 1540   ∈ wcel 2108

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-cleq 2727  df-clel 2809

This theorem is referenced by:  oancom  9665  0r  11094  1sr  11095  m1r  11096  smndex1ibas  18878  recvs  25097  recvsOLD  25098  qcvs  25099  wlk2v2elem1  30136  konigsbergiedgw  30229  lmxrge0  33983  brsigarn  34215  ex-sategoelel12  35449  sinccvglem  35694  bj-minftyccb  37243  resuppsinopn  42406  omcl3g  43358  fouriersw  46260