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Mathbox for Wolf Lammen |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > wl-3xorbi | Structured version Visualization version GIF version |
Description: Triple xor can be replaced with a triple biconditional. Unlike ⊻, you cannot add more inputs by simply stacking up more biconditionals, and still express an "odd number of inputs". (Contributed by Wolf Lammen, 24-Apr-2024.) |
Ref | Expression |
---|---|
wl-3xorbi | ⊢ (hadd(𝜑, 𝜓, 𝜒) ↔ (𝜑 ↔ (𝜓 ↔ 𝜒))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wl-df3xor2 35151 | . 2 ⊢ (hadd(𝜑, 𝜓, 𝜒) ↔ (𝜑 ⊻ (𝜓 ⊻ 𝜒))) | |
2 | df-xor 1504 | . 2 ⊢ ((𝜑 ⊻ (𝜓 ⊻ 𝜒)) ↔ ¬ (𝜑 ↔ (𝜓 ⊻ 𝜒))) | |
3 | xor3 388 | . . 3 ⊢ (¬ (𝜑 ↔ (𝜓 ⊻ 𝜒)) ↔ (𝜑 ↔ ¬ (𝜓 ⊻ 𝜒))) | |
4 | xnor 1505 | . . . 4 ⊢ ((𝜓 ↔ 𝜒) ↔ ¬ (𝜓 ⊻ 𝜒)) | |
5 | 4 | bibi2i 342 | . . 3 ⊢ ((𝜑 ↔ (𝜓 ↔ 𝜒)) ↔ (𝜑 ↔ ¬ (𝜓 ⊻ 𝜒))) |
6 | 3, 5 | bitr4i 281 | . 2 ⊢ (¬ (𝜑 ↔ (𝜓 ⊻ 𝜒)) ↔ (𝜑 ↔ (𝜓 ↔ 𝜒))) |
7 | 1, 2, 6 | 3bitri 301 | 1 ⊢ (hadd(𝜑, 𝜓, 𝜒) ↔ (𝜑 ↔ (𝜓 ↔ 𝜒))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 209 ⊻ wxo 1503 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-3xorbi2 35156 wl-3xorrot 35159 wl-3xornot1 35162 |
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