Theorem symdifass 4207

Description: Symmetric difference is associative. (Contributed by Scott Fenton, 24-Apr-2012.) (Proof shortened by BJ, 7-Sep-2022.)

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		elsymdifxor 4205	. . 3 ⊢ (𝑥 ∈ ((𝐴 △ 𝐵) △ 𝐶) ↔ (𝑥 ∈ (𝐴 △ 𝐵) ⊻ 𝑥 ∈ 𝐶))
2		elsymdifxor 4205	. . . . 5 ⊢ (𝑥 ∈ (𝐴 △ 𝐵) ↔ (𝑥 ∈ 𝐴 ⊻ 𝑥 ∈ 𝐵))
3		biid 264	. . . . 5 ⊢ (𝑥 ∈ 𝐶 ↔ 𝑥 ∈ 𝐶)
4	2, 3	xorbi12i 1553	. . . 4 ⊢ ((𝑥 ∈ (𝐴 △ 𝐵) ⊻ 𝑥 ∈ 𝐶) ↔ ((𝑥 ∈ 𝐴 ⊻ 𝑥 ∈ 𝐵) ⊻ 𝑥 ∈ 𝐶))
5		xorass 1544	. . . 4 ⊢ (((𝑥 ∈ 𝐴 ⊻ 𝑥 ∈ 𝐵) ⊻ 𝑥 ∈ 𝐶) ↔ (𝑥 ∈ 𝐴 ⊻ (𝑥 ∈ 𝐵 ⊻ 𝑥 ∈ 𝐶)))
6		biid 264	. . . . 5 ⊢ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴)
7		elsymdifxor 4205	. . . . . 6 ⊢ (𝑥 ∈ (𝐵 △ 𝐶) ↔ (𝑥 ∈ 𝐵 ⊻ 𝑥 ∈ 𝐶))
8	7	bicomi 227	. . . . 5 ⊢ ((𝑥 ∈ 𝐵 ⊻ 𝑥 ∈ 𝐶) ↔ 𝑥 ∈ (𝐵 △ 𝐶))
9	6, 8	xorbi12i 1553	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ⊻ (𝑥 ∈ 𝐵 ⊻ 𝑥 ∈ 𝐶)) ↔ (𝑥 ∈ 𝐴 ⊻ 𝑥 ∈ (𝐵 △ 𝐶)))
10	4, 5, 9	3bitri 300	. . 3 ⊢ ((𝑥 ∈ (𝐴 △ 𝐵) ⊻ 𝑥 ∈ 𝐶) ↔ (𝑥 ∈ 𝐴 ⊻ 𝑥 ∈ (𝐵 △ 𝐶)))
11		elsymdifxor 4205	. . . 4 ⊢ (𝑥 ∈ (𝐴 △ (𝐵 △ 𝐶)) ↔ (𝑥 ∈ 𝐴 ⊻ 𝑥 ∈ (𝐵 △ 𝐶)))
12	11	bicomi 227	. . 3 ⊢ ((𝑥 ∈ 𝐴 ⊻ 𝑥 ∈ (𝐵 △ 𝐶)) ↔ 𝑥 ∈ (𝐴 △ (𝐵 △ 𝐶)))
13	1, 10, 12	3bitri 300	. 2 ⊢ (𝑥 ∈ ((𝐴 △ 𝐵) △ 𝐶) ↔ 𝑥 ∈ (𝐴 △ (𝐵 △ 𝐶)))
14	13	eqriv 2757	1 ⊢ ((𝐴 △ 𝐵) △ 𝐶) = (𝐴 △ (𝐵 △ 𝐶))

This proof depends on syntax axioms:   ⊻ wxo 1540   = wceq 1569   ∈ wcel 2144   △ csymdif 4197

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-xor 1541  df-tru 1572  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-v 3452  df-dif 4047  df-un 4151  df-symdif 4198

This theorem is used by: (None)