Theorem 3eltr4i 2701

Description: Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.)

Hypotheses

Ref	Expression
3eltr4.1	⊢ 𝐴 ∈ 𝐵
3eltr4.2	⊢ 𝐶 = 𝐴
3eltr4.3	⊢ 𝐷 = 𝐵

Assertion

Ref	Expression
3eltr4i	⊢ 𝐶 ∈ 𝐷

Proof of Theorem 3eltr4i

Step	Hyp	Ref	Expression
1		3eltr4.2	. 2 ⊢ 𝐶 = 𝐴
2		3eltr4.1	. . 3 ⊢ 𝐴 ∈ 𝐵
3		3eltr4.3	. . 3 ⊢ 𝐷 = 𝐵
4	2, 3	eleqtrri 2687	. 2 ⊢ 𝐴 ∈ 𝐷
5	1, 4	eqeltri 2684	1 ⊢ 𝐶 ∈ 𝐷

Syntax hints:   = wceq 1475   ∈ wcel 1977

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696  df-cleq 2603  df-clel 2606

This theorem is referenced by:  oancom  8431  0r  9780  1sr  9781  m1r  9782  recvs  22754  qcvs  22755  lmxrge0  29326  brsigarn  29574  sinccvglem  30820  bj-minftyccb  32289  fouriersw  39124  1wlk2v2elem1  41322  konigsbergiedgw  41416  konigsbergiedgwOLD  41417