Theorem symdifass 3853

Description: Symmetric difference associates. (Contributed by Scott Fenton, 24-Apr-2012.)

Assertion

Ref	Expression
symdifass	⊢ (𝐴 △ (𝐵 △ 𝐶)) = ((𝐴 △ 𝐵) △ 𝐶)

Proof of Theorem symdifass

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		biass 374	. . . . . . 7 ⊢ (((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ↔ 𝑥 ∈ 𝐶) ↔ (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐵 ↔ 𝑥 ∈ 𝐶)))
2	1	notbii 310	. . . . . 6 ⊢ (¬ ((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ↔ 𝑥 ∈ 𝐶) ↔ ¬ (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐵 ↔ 𝑥 ∈ 𝐶)))
3		xor3 372	. . . . . . . 8 ⊢ (¬ (¬ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ↔ 𝑥 ∈ 𝐶) ↔ (¬ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ↔ ¬ 𝑥 ∈ 𝐶))
4		notbi 309	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ↔ 𝑥 ∈ 𝐶) ↔ (¬ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ↔ ¬ 𝑥 ∈ 𝐶))
5	3, 4	bitr4i 267	. . . . . . 7 ⊢ (¬ (¬ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ↔ 𝑥 ∈ 𝐶) ↔ ((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ↔ 𝑥 ∈ 𝐶))
6	5	con1bii 346	. . . . . 6 ⊢ (¬ ((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ↔ 𝑥 ∈ 𝐶) ↔ (¬ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ↔ 𝑥 ∈ 𝐶))
7		xor3 372	. . . . . 6 ⊢ (¬ (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐵 ↔ 𝑥 ∈ 𝐶)) ↔ (𝑥 ∈ 𝐴 ↔ ¬ (𝑥 ∈ 𝐵 ↔ 𝑥 ∈ 𝐶)))
8	2, 6, 7	3bitr3ri 291	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ↔ ¬ (𝑥 ∈ 𝐵 ↔ 𝑥 ∈ 𝐶)) ↔ (¬ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ↔ 𝑥 ∈ 𝐶))
9		elsymdif 3849	. . . . . 6 ⊢ (𝑥 ∈ (𝐵 △ 𝐶) ↔ ¬ (𝑥 ∈ 𝐵 ↔ 𝑥 ∈ 𝐶))
10	9	bibi2i 327	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ (𝐵 △ 𝐶)) ↔ (𝑥 ∈ 𝐴 ↔ ¬ (𝑥 ∈ 𝐵 ↔ 𝑥 ∈ 𝐶)))
11		elsymdif 3849	. . . . . 6 ⊢ (𝑥 ∈ (𝐴 △ 𝐵) ↔ ¬ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))
12	11	bibi1i 328	. . . . 5 ⊢ ((𝑥 ∈ (𝐴 △ 𝐵) ↔ 𝑥 ∈ 𝐶) ↔ (¬ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ↔ 𝑥 ∈ 𝐶))
13	8, 10, 12	3bitr4i 292	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ (𝐵 △ 𝐶)) ↔ (𝑥 ∈ (𝐴 △ 𝐵) ↔ 𝑥 ∈ 𝐶))
14	13	notbii 310	. . 3 ⊢ (¬ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ (𝐵 △ 𝐶)) ↔ ¬ (𝑥 ∈ (𝐴 △ 𝐵) ↔ 𝑥 ∈ 𝐶))
15		elsymdif 3849	. . 3 ⊢ (𝑥 ∈ (𝐴 △ (𝐵 △ 𝐶)) ↔ ¬ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ (𝐵 △ 𝐶)))
16		elsymdif 3849	. . 3 ⊢ (𝑥 ∈ ((𝐴 △ 𝐵) △ 𝐶) ↔ ¬ (𝑥 ∈ (𝐴 △ 𝐵) ↔ 𝑥 ∈ 𝐶))
17	14, 15, 16	3bitr4i 292	. 2 ⊢ (𝑥 ∈ (𝐴 △ (𝐵 △ 𝐶)) ↔ 𝑥 ∈ ((𝐴 △ 𝐵) △ 𝐶))
18	17	eqriv 2619	1 ⊢ (𝐴 △ (𝐵 △ 𝐶)) = ((𝐴 △ 𝐵) △ 𝐶)

Syntax hints:  ¬ wn 3   ↔ wb 196   = wceq 1483   ∈ wcel 1990   △ csymdif 3843

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-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-dif 3577  df-un 3579  df-symdif 3844

This theorem is referenced by: (None)