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Mirrors > Home > MPE Home > Th. List > symdifass | Structured version Visualization version GIF version |
Description: Symmetric difference is associative. (Contributed by Scott Fenton, 24-Apr-2012.) (Proof shortened by BJ, 7-Sep-2022.) |
Ref | Expression |
---|---|
symdifass | ⊢ ((𝐴 △ 𝐵) △ 𝐶) = (𝐴 △ (𝐵 △ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elsymdifxor 4228 | . . 3 ⊢ (𝑥 ∈ ((𝐴 △ 𝐵) △ 𝐶) ↔ (𝑥 ∈ (𝐴 △ 𝐵) ⊻ 𝑥 ∈ 𝐶)) | |
2 | elsymdifxor 4228 | . . . . 5 ⊢ (𝑥 ∈ (𝐴 △ 𝐵) ↔ (𝑥 ∈ 𝐴 ⊻ 𝑥 ∈ 𝐵)) | |
3 | biid 263 | . . . . 5 ⊢ (𝑥 ∈ 𝐶 ↔ 𝑥 ∈ 𝐶) | |
4 | 2, 3 | xorbi12i 1515 | . . . 4 ⊢ ((𝑥 ∈ (𝐴 △ 𝐵) ⊻ 𝑥 ∈ 𝐶) ↔ ((𝑥 ∈ 𝐴 ⊻ 𝑥 ∈ 𝐵) ⊻ 𝑥 ∈ 𝐶)) |
5 | xorass 1506 | . . . 4 ⊢ (((𝑥 ∈ 𝐴 ⊻ 𝑥 ∈ 𝐵) ⊻ 𝑥 ∈ 𝐶) ↔ (𝑥 ∈ 𝐴 ⊻ (𝑥 ∈ 𝐵 ⊻ 𝑥 ∈ 𝐶))) | |
6 | biid 263 | . . . . 5 ⊢ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴) | |
7 | elsymdifxor 4228 | . . . . . 6 ⊢ (𝑥 ∈ (𝐵 △ 𝐶) ↔ (𝑥 ∈ 𝐵 ⊻ 𝑥 ∈ 𝐶)) | |
8 | 7 | bicomi 226 | . . . . 5 ⊢ ((𝑥 ∈ 𝐵 ⊻ 𝑥 ∈ 𝐶) ↔ 𝑥 ∈ (𝐵 △ 𝐶)) |
9 | 6, 8 | xorbi12i 1515 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 ⊻ (𝑥 ∈ 𝐵 ⊻ 𝑥 ∈ 𝐶)) ↔ (𝑥 ∈ 𝐴 ⊻ 𝑥 ∈ (𝐵 △ 𝐶))) |
10 | 4, 5, 9 | 3bitri 299 | . . 3 ⊢ ((𝑥 ∈ (𝐴 △ 𝐵) ⊻ 𝑥 ∈ 𝐶) ↔ (𝑥 ∈ 𝐴 ⊻ 𝑥 ∈ (𝐵 △ 𝐶))) |
11 | elsymdifxor 4228 | . . . 4 ⊢ (𝑥 ∈ (𝐴 △ (𝐵 △ 𝐶)) ↔ (𝑥 ∈ 𝐴 ⊻ 𝑥 ∈ (𝐵 △ 𝐶))) | |
12 | 11 | bicomi 226 | . . 3 ⊢ ((𝑥 ∈ 𝐴 ⊻ 𝑥 ∈ (𝐵 △ 𝐶)) ↔ 𝑥 ∈ (𝐴 △ (𝐵 △ 𝐶))) |
13 | 1, 10, 12 | 3bitri 299 | . 2 ⊢ (𝑥 ∈ ((𝐴 △ 𝐵) △ 𝐶) ↔ 𝑥 ∈ (𝐴 △ (𝐵 △ 𝐶))) |
14 | 13 | eqriv 2820 | 1 ⊢ ((𝐴 △ 𝐵) △ 𝐶) = (𝐴 △ (𝐵 △ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: ⊻ wxo 1501 = wceq 1537 ∈ wcel 2114 △ csymdif 4220 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-xor 1502 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-v 3498 df-dif 3941 df-un 3943 df-symdif 4221 |
This theorem is referenced by: (None) |
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