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Theorem symdifass 3886
Description: Symmetric difference associates. (Contributed by Scott Fenton, 24-Apr-2012.)
Assertion
Ref Expression
symdifass (𝐴 △ (𝐵𝐶)) = ((𝐴𝐵) △ 𝐶)

Proof of Theorem symdifass
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 biass 373 . . . . . . 7 (((𝑥𝐴𝑥𝐵) ↔ 𝑥𝐶) ↔ (𝑥𝐴 ↔ (𝑥𝐵𝑥𝐶)))
21notbii 309 . . . . . 6 (¬ ((𝑥𝐴𝑥𝐵) ↔ 𝑥𝐶) ↔ ¬ (𝑥𝐴 ↔ (𝑥𝐵𝑥𝐶)))
3 xor3 371 . . . . . . . 8 (¬ (¬ (𝑥𝐴𝑥𝐵) ↔ 𝑥𝐶) ↔ (¬ (𝑥𝐴𝑥𝐵) ↔ ¬ 𝑥𝐶))
4 notbi 308 . . . . . . . 8 (((𝑥𝐴𝑥𝐵) ↔ 𝑥𝐶) ↔ (¬ (𝑥𝐴𝑥𝐵) ↔ ¬ 𝑥𝐶))
53, 4bitr4i 267 . . . . . . 7 (¬ (¬ (𝑥𝐴𝑥𝐵) ↔ 𝑥𝐶) ↔ ((𝑥𝐴𝑥𝐵) ↔ 𝑥𝐶))
65con1bii 345 . . . . . 6 (¬ ((𝑥𝐴𝑥𝐵) ↔ 𝑥𝐶) ↔ (¬ (𝑥𝐴𝑥𝐵) ↔ 𝑥𝐶))
7 xor3 371 . . . . . 6 (¬ (𝑥𝐴 ↔ (𝑥𝐵𝑥𝐶)) ↔ (𝑥𝐴 ↔ ¬ (𝑥𝐵𝑥𝐶)))
82, 6, 73bitr3ri 291 . . . . 5 ((𝑥𝐴 ↔ ¬ (𝑥𝐵𝑥𝐶)) ↔ (¬ (𝑥𝐴𝑥𝐵) ↔ 𝑥𝐶))
9 elsymdif 3882 . . . . . 6 (𝑥 ∈ (𝐵𝐶) ↔ ¬ (𝑥𝐵𝑥𝐶))
109bibi2i 326 . . . . 5 ((𝑥𝐴𝑥 ∈ (𝐵𝐶)) ↔ (𝑥𝐴 ↔ ¬ (𝑥𝐵𝑥𝐶)))
11 elsymdif 3882 . . . . . 6 (𝑥 ∈ (𝐴𝐵) ↔ ¬ (𝑥𝐴𝑥𝐵))
1211bibi1i 327 . . . . 5 ((𝑥 ∈ (𝐴𝐵) ↔ 𝑥𝐶) ↔ (¬ (𝑥𝐴𝑥𝐵) ↔ 𝑥𝐶))
138, 10, 123bitr4i 292 . . . 4 ((𝑥𝐴𝑥 ∈ (𝐵𝐶)) ↔ (𝑥 ∈ (𝐴𝐵) ↔ 𝑥𝐶))
1413notbii 309 . . 3 (¬ (𝑥𝐴𝑥 ∈ (𝐵𝐶)) ↔ ¬ (𝑥 ∈ (𝐴𝐵) ↔ 𝑥𝐶))
15 elsymdif 3882 . . 3 (𝑥 ∈ (𝐴 △ (𝐵𝐶)) ↔ ¬ (𝑥𝐴𝑥 ∈ (𝐵𝐶)))
16 elsymdif 3882 . . 3 (𝑥 ∈ ((𝐴𝐵) △ 𝐶) ↔ ¬ (𝑥 ∈ (𝐴𝐵) ↔ 𝑥𝐶))
1714, 15, 163bitr4i 292 . 2 (𝑥 ∈ (𝐴 △ (𝐵𝐶)) ↔ 𝑥 ∈ ((𝐴𝐵) △ 𝐶))
1817eqriv 2648 1 (𝐴 △ (𝐵𝐶)) = ((𝐴𝐵) △ 𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196   = wceq 1523  wcel 2030  csymdif 3876
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-v 3233  df-dif 3610  df-un 3612  df-symdif 3877
This theorem is referenced by: (None)
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