Theorem 3eltr4i 2851

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 2837	. 2 ⊢ 𝐴 ∈ 𝐷
5	1, 4	eqeltri 2834	1 ⊢ 𝐶 ∈ 𝐷

Syntax hints:   = wceq 1540   ∈ wcel 2105

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2708

This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1781  df-cleq 2729  df-clel 2815

This theorem is referenced by:  oancom  9477  0r  10906  1sr  10907  m1r  10908  smndex1ibas  18606  recvs  24380  recvsOLD  24381  qcvs  24382  wlk2v2elem1  28627  konigsbergiedgw  28720  lmxrge0  32008  brsigarn  32258  ex-sategoelel12  33495  sinccvglem  33736  bj-minftyccb  35456  fouriersw  44016