Theorem 3eltr3i 2874

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

Hypotheses

Ref	Expression
3eltr3i.1	⊢ 𝐴 ∈ 𝐵
3eltr3i.2	⊢ 𝐴 = 𝐶
3eltr3i.3	⊢ 𝐵 = 𝐷

Assertion

Ref	Expression
3eltr3i	⊢ 𝐶 ∈ 𝐷

Proof of Theorem 3eltr3i

Step	Hyp	Ref	Expression
1		3eltr3i.2	. 2 ⊢ 𝐴 = 𝐶
2		3eltr3i.1	. . 3 ⊢ 𝐴 ∈ 𝐵
3		3eltr3i.3	. . 3 ⊢ 𝐵 = 𝐷
4	2, 3	eleqtri 2860	. 2 ⊢ 𝐴 ∈ 𝐷
5	1, 4	eqeltrri 2859	1 ⊢ 𝐶 ∈ 𝐷

This proof depends on syntax axioms:   = wceq 1570   ∈ wcel 2145

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-cleq 2754  df-clel 2837

This theorem is used by:  raddcn  34426  clsk1independent  44873  fourierdlem62  46983