Theorem 3eltr4i 2852

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 2838	. 2 ⊢ 𝐴 ∈ 𝐷
5	1, 4	eqeltri 2835	1 ⊢ 𝐶 ∈ 𝐷

Syntax hints:   = wceq 1539   ∈ wcel 2108

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1784  df-cleq 2730  df-clel 2817

This theorem is referenced by:  oancom  9339  0r  10767  1sr  10768  m1r  10769  smndex1ibas  18454  recvs  24215  qcvs  24216  wlk2v2elem1  28420  konigsbergiedgw  28513  lmxrge0  31804  brsigarn  32052  ex-sategoelel12  33289  sinccvglem  33530  bj-minftyccb  35323  fouriersw  43662