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 1536   ∈ wcel 2105

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1776  df-cleq 2726  df-clel 2813

This theorem is referenced by:  oancom  9688  0r  11117  1sr  11118  m1r  11119  smndex1ibas  18925  recvs  25192  recvsOLD  25193  qcvs  25194  wlk2v2elem1  30183  konigsbergiedgw  30276  lmxrge0  33912  brsigarn  34164  ex-sategoelel12  35411  sinccvglem  35656  bj-minftyccb  37207  omcl3g  43323  fouriersw  46186