Theorem 3eltr4i 2714

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 2700	. 2 ⊢ 𝐴 ∈ 𝐷
5	1, 4	eqeltri 2697	1 ⊢ 𝐶 ∈ 𝐷

Syntax hints:   = wceq 1483   ∈ wcel 1990

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705  df-cleq 2615  df-clel 2618

This theorem is referenced by:  oancom  8548  0r  9901  1sr  9902  m1r  9903  recvs  22946  qcvs  22947  wlk2v2elem1  27015  konigsbergiedgw  27108  konigsbergiedgwOLD  27109  lmxrge0  29998  brsigarn  30247  sinccvglem  31566  bj-minftyccb  33112  fouriersw  40448