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Mathbox for Wolf Lammen |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > wl-3xorrot | Structured version Visualization version GIF version |
Description: Rotation law for triple xor. (Contributed by Mario Carneiro, 4-Sep-2016.) df-had redefined. (Revised by Wolf Lammen, 24-Apr-2024.) |
Ref | Expression |
---|---|
wl-3xorrot | ⊢ (hadd(𝜑, 𝜓, 𝜒) ↔ hadd(𝜓, 𝜒, 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bicom 225 | . 2 ⊢ ((𝜑 ↔ (𝜓 ↔ 𝜒)) ↔ ((𝜓 ↔ 𝜒) ↔ 𝜑)) | |
2 | wl-3xorbi 35155 | . 2 ⊢ (hadd(𝜑, 𝜓, 𝜒) ↔ (𝜑 ↔ (𝜓 ↔ 𝜒))) | |
3 | wl-3xorbi2 35156 | . 2 ⊢ (hadd(𝜓, 𝜒, 𝜑) ↔ ((𝜓 ↔ 𝜒) ↔ 𝜑)) | |
4 | 1, 2, 3 | 3bitr4i 307 | 1 ⊢ (hadd(𝜑, 𝜓, 𝜒) ↔ hadd(𝜓, 𝜒, 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 haddwhad 1595 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 |
This theorem depends on definitions: df-bi 210 df-an 401 df-or 846 df-ifp 1060 df-xor 1504 df-tru 1542 df-had 1596 |
This theorem is referenced by: wl-3xorcomb 35161 |
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