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Theorem symdifass 4211
 Description: Symmetric difference is associative. (Contributed by Scott Fenton, 24-Apr-2012.) (Proof shortened by BJ, 7-Sep-2022.)
Assertion
Ref Expression
symdifass ((𝐴𝐵) △ 𝐶) = (𝐴 △ (𝐵𝐶))

Proof of Theorem symdifass
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 elsymdifxor 4209 . . 3 (𝑥 ∈ ((𝐴𝐵) △ 𝐶) ↔ (𝑥 ∈ (𝐴𝐵) ⊻ 𝑥𝐶))
2 elsymdifxor 4209 . . . . 5 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
3 biid 264 . . . . 5 (𝑥𝐶𝑥𝐶)
42, 3xorbi12i 1516 . . . 4 ((𝑥 ∈ (𝐴𝐵) ⊻ 𝑥𝐶) ↔ ((𝑥𝐴𝑥𝐵) ⊻ 𝑥𝐶))
5 xorass 1507 . . . 4 (((𝑥𝐴𝑥𝐵) ⊻ 𝑥𝐶) ↔ (𝑥𝐴 ⊻ (𝑥𝐵𝑥𝐶)))
6 biid 264 . . . . 5 (𝑥𝐴𝑥𝐴)
7 elsymdifxor 4209 . . . . . 6 (𝑥 ∈ (𝐵𝐶) ↔ (𝑥𝐵𝑥𝐶))
87bicomi 227 . . . . 5 ((𝑥𝐵𝑥𝐶) ↔ 𝑥 ∈ (𝐵𝐶))
96, 8xorbi12i 1516 . . . 4 ((𝑥𝐴 ⊻ (𝑥𝐵𝑥𝐶)) ↔ (𝑥𝐴𝑥 ∈ (𝐵𝐶)))
104, 5, 93bitri 300 . . 3 ((𝑥 ∈ (𝐴𝐵) ⊻ 𝑥𝐶) ↔ (𝑥𝐴𝑥 ∈ (𝐵𝐶)))
11 elsymdifxor 4209 . . . 4 (𝑥 ∈ (𝐴 △ (𝐵𝐶)) ↔ (𝑥𝐴𝑥 ∈ (𝐵𝐶)))
1211bicomi 227 . . 3 ((𝑥𝐴𝑥 ∈ (𝐵𝐶)) ↔ 𝑥 ∈ (𝐴 △ (𝐵𝐶)))
131, 10, 123bitri 300 . 2 (𝑥 ∈ ((𝐴𝐵) △ 𝐶) ↔ 𝑥 ∈ (𝐴 △ (𝐵𝐶)))
1413eqriv 2821 1 ((𝐴𝐵) △ 𝐶) = (𝐴 △ (𝐵𝐶))
 Colors of variables: wff setvar class Syntax hints:   ⊻ wxo 1502   = wceq 1538   ∈ wcel 2115   △ csymdif 4201 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-ext 2796 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-xor 1503  df-ex 1782  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-v 3481  df-dif 3921  df-un 3923  df-symdif 4202 This theorem is referenced by: (None)
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