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Theorem symdifass 4081
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 4079 . . 3 (𝑥 ∈ ((𝐴𝐵) △ 𝐶) ↔ (𝑥 ∈ (𝐴𝐵) ⊻ 𝑥𝐶))
2 elsymdifxor 4079 . . . . 5 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
3 biid 253 . . . . 5 (𝑥𝐶𝑥𝐶)
42, 3xorbi12i 1650 . . . 4 ((𝑥 ∈ (𝐴𝐵) ⊻ 𝑥𝐶) ↔ ((𝑥𝐴𝑥𝐵) ⊻ 𝑥𝐶))
5 xorass 1641 . . . 4 (((𝑥𝐴𝑥𝐵) ⊻ 𝑥𝐶) ↔ (𝑥𝐴 ⊻ (𝑥𝐵𝑥𝐶)))
6 biid 253 . . . . 5 (𝑥𝐴𝑥𝐴)
7 elsymdifxor 4079 . . . . . 6 (𝑥 ∈ (𝐵𝐶) ↔ (𝑥𝐵𝑥𝐶))
87bicomi 216 . . . . 5 ((𝑥𝐵𝑥𝐶) ↔ 𝑥 ∈ (𝐵𝐶))
96, 8xorbi12i 1650 . . . 4 ((𝑥𝐴 ⊻ (𝑥𝐵𝑥𝐶)) ↔ (𝑥𝐴𝑥 ∈ (𝐵𝐶)))
104, 5, 93bitri 289 . . 3 ((𝑥 ∈ (𝐴𝐵) ⊻ 𝑥𝐶) ↔ (𝑥𝐴𝑥 ∈ (𝐵𝐶)))
11 elsymdifxor 4079 . . . 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 2822 1 ((𝐴𝐵) △ 𝐶) = (𝐴 △ (𝐵𝐶))
Colors of variables: wff setvar class
Syntax hints:  wxo 1637   = wceq 1656  wcel 2164  csymdif 4071
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-ext 2803
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-xor 1638  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-v 3416  df-dif 3801  df-un 3803  df-symdif 4072
This theorem is referenced by: (None)
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