Theorem 3eltr3i 2927

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 2913	. 2 ⊢ 𝐴 ∈ 𝐷
5	1, 4	eqeltrri 2912	1 ⊢ 𝐶 ∈ 𝐷

Syntax hints:   = wceq 1537   ∈ wcel 2114

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-ext 2795

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1781  df-cleq 2816  df-clel 2895

This theorem is referenced by:  raddcn  31174  clsk1independent  40403  fourierdlem62  42460