Theorem 3eltr3i 2713

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

Hypotheses

Ref	Expression
3eltr3.1	⊢ 𝐴 ∈ 𝐵
3eltr3.2	⊢ 𝐴 = 𝐶
3eltr3.3	⊢ 𝐵 = 𝐷

Assertion

Ref	Expression
3eltr3i	⊢ 𝐶 ∈ 𝐷

Proof of Theorem 3eltr3i

Step	Hyp	Ref	Expression
1		3eltr3.2	. 2 ⊢ 𝐴 = 𝐶
2		3eltr3.1	. . 3 ⊢ 𝐴 ∈ 𝐵
3		3eltr3.3	. . 3 ⊢ 𝐵 = 𝐷
4	2, 3	eleqtri 2699	. 2 ⊢ 𝐴 ∈ 𝐷
5	1, 4	eqeltrri 2698	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:  raddcn  29975  clsk1independent  38344  fourierdlem62  40385