Theorem 3eltr4i 2320

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 2314	. 2 ⊢ 𝐴 ∈ 𝐷
5	1, 4	eqeltri 2311	1 ⊢ 𝐶 ∈ 𝐷

Syntax hints:   = wceq 1402   ∈ wcel 2209

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-4 1563  ax-17 1579  ax-ial 1587  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-cleq 2231  df-clel 2234

This theorem is referenced by:  1nq  7723  0r  8107  1sr  8108  m1r  8109  konigsbergiedgwen  16639