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Theorem symdifass 4050
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 4048 . . 3 (𝑥 ∈ ((𝐴𝐵) △ 𝐶) ↔ (𝑥 ∈ (𝐴𝐵) ⊻ 𝑥𝐶))
2 elsymdifxor 4048 . . . . 5 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
3 biid 253 . . . . 5 (𝑥𝐶𝑥𝐶)
42, 3xorbi12i 1647 . . . 4 ((𝑥 ∈ (𝐴𝐵) ⊻ 𝑥𝐶) ↔ ((𝑥𝐴𝑥𝐵) ⊻ 𝑥𝐶))
5 xorass 1638 . . . 4 (((𝑥𝐴𝑥𝐵) ⊻ 𝑥𝐶) ↔ (𝑥𝐴 ⊻ (𝑥𝐵𝑥𝐶)))
6 biid 253 . . . . 5 (𝑥𝐴𝑥𝐴)
7 elsymdifxor 4048 . . . . . 6 (𝑥 ∈ (𝐵𝐶) ↔ (𝑥𝐵𝑥𝐶))
87bicomi 216 . . . . 5 ((𝑥𝐵𝑥𝐶) ↔ 𝑥 ∈ (𝐵𝐶))
96, 8xorbi12i 1647 . . . 4 ((𝑥𝐴 ⊻ (𝑥𝐵𝑥𝐶)) ↔ (𝑥𝐴𝑥 ∈ (𝐵𝐶)))
104, 5, 93bitri 289 . . 3 ((𝑥 ∈ (𝐴𝐵) ⊻ 𝑥𝐶) ↔ (𝑥𝐴𝑥 ∈ (𝐵𝐶)))
11 elsymdifxor 4048 . . . 4 (𝑥 ∈ (𝐴 △ (𝐵𝐶)) ↔ (𝑥𝐴𝑥 ∈ (𝐵𝐶)))
1211bicomi 216 . . 3 ((𝑥𝐴𝑥 ∈ (𝐵𝐶)) ↔ 𝑥 ∈ (𝐴 △ (𝐵𝐶)))
131, 10, 123bitri 289 . 2 (𝑥 ∈ ((𝐴𝐵) △ 𝐶) ↔ 𝑥 ∈ (𝐴 △ (𝐵𝐶)))
1413eqriv 2796 1 ((𝐴𝐵) △ 𝐶) = (𝐴 △ (𝐵𝐶))
Colors of variables: wff setvar class
Syntax hints:  wxo 1634   = wceq 1653  wcel 2157  csymdif 4040
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-ext 2777
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-xor 1635  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-v 3387  df-dif 3772  df-un 3774  df-symdif 4041
This theorem is referenced by: (None)
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