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| Mirrors > Home > MPE Home > Th. List > Mathboxes > wl-df3maxtru1 | Structured version Visualization version GIF version | ||
| Description: Assuming
"(n+1)-maxtru1" ↔ ¬
"(n+1)-mintru-2", we can deduce from
     the recursion formula given in wl-df-3mintru2 37485, that a similiar one "(n+1)-maxtru1" ↔ if-(𝜑,-. "n-mintru-1" , "n-maxtru1" ) is valid for expressing 'at most one input is true'. This can also be rephrased as a mutual exclusivity of propositional expressions (no two of a sequence of inputs can simultaneously be true). Of course, this suggests that all inputs depend on variables 𝜂, 𝜁... Whatever wellformed expression we plugin for these variables, it will render at most one of the inputs true. The here introduced mutual exclusivity is possibly useful for case studies, where we want the cases be sort of 'disjoint'. One can further imagine that a complete case scenario demands that the 'at most' is sharpened to 'exactly one'. This does not impose any difficulty here, as one of the inputs will then be the negation of all others be or'ed. As one input is determined, 'at most one' is sufficient to describe the general form here. Since cadd is an alias for 'at least 2 out of three are true', its negation is under focus here. (Contributed by Wolf Lammen, 23-Jun-2024.) | 
| Ref | Expression | 
|---|---|
| wl-df3maxtru1 | ⊢ (¬ cadd(𝜑, 𝜓, 𝜒) ↔ if-(𝜑, (𝜓 ⊽ 𝜒), (𝜓 ⊼ 𝜒))) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | cadnot 1615 | . 2 ⊢ (¬ cadd(𝜑, 𝜓, 𝜒) ↔ cadd(¬ 𝜑, ¬ 𝜓, ¬ 𝜒)) | |
| 2 | wl-df-3mintru2 37485 | . 2 ⊢ (cadd(¬ 𝜑, ¬ 𝜓, ¬ 𝜒) ↔ if-(¬ 𝜑, (¬ 𝜓 ∨ ¬ 𝜒), (¬ 𝜓 ∧ ¬ 𝜒))) | |
| 3 | ifpn 1074 | . . 3 ⊢ (if-(𝜑, (𝜓 ⊽ 𝜒), (𝜓 ⊼ 𝜒)) ↔ if-(¬ 𝜑, (𝜓 ⊼ 𝜒), (𝜓 ⊽ 𝜒))) | |
| 4 | nanor 1495 | . . . . . 6 ⊢ ((𝜓 ⊼ 𝜒) ↔ (¬ 𝜓 ∨ ¬ 𝜒)) | |
| 5 | 4 | a1i 11 | . . . . 5 ⊢ (⊤ → ((𝜓 ⊼ 𝜒) ↔ (¬ 𝜓 ∨ ¬ 𝜒))) | 
| 6 | df-nor 1529 | . . . . . . 7 ⊢ ((𝜓 ⊽ 𝜒) ↔ ¬ (𝜓 ∨ 𝜒)) | |
| 7 | ioran 986 | . . . . . . 7 ⊢ (¬ (𝜓 ∨ 𝜒) ↔ (¬ 𝜓 ∧ ¬ 𝜒)) | |
| 8 | 6, 7 | bitri 275 | . . . . . 6 ⊢ ((𝜓 ⊽ 𝜒) ↔ (¬ 𝜓 ∧ ¬ 𝜒)) | 
| 9 | 8 | a1i 11 | . . . . 5 ⊢ (⊤ → ((𝜓 ⊽ 𝜒) ↔ (¬ 𝜓 ∧ ¬ 𝜒))) | 
| 10 | 5, 9 | ifpbi23d 1080 | . . . 4 ⊢ (⊤ → (if-(¬ 𝜑, (𝜓 ⊼ 𝜒), (𝜓 ⊽ 𝜒)) ↔ if-(¬ 𝜑, (¬ 𝜓 ∨ ¬ 𝜒), (¬ 𝜓 ∧ ¬ 𝜒)))) | 
| 11 | 10 | mptru 1547 | . . 3 ⊢ (if-(¬ 𝜑, (𝜓 ⊼ 𝜒), (𝜓 ⊽ 𝜒)) ↔ if-(¬ 𝜑, (¬ 𝜓 ∨ ¬ 𝜒), (¬ 𝜓 ∧ ¬ 𝜒))) | 
| 12 | 3, 11 | bitr2i 276 | . 2 ⊢ (if-(¬ 𝜑, (¬ 𝜓 ∨ ¬ 𝜒), (¬ 𝜓 ∧ ¬ 𝜒)) ↔ if-(𝜑, (𝜓 ⊽ 𝜒), (𝜓 ⊼ 𝜒))) | 
| 13 | 1, 2, 12 | 3bitri 297 | 1 ⊢ (¬ cadd(𝜑, 𝜓, 𝜒) ↔ if-(𝜑, (𝜓 ⊽ 𝜒), (𝜓 ⊼ 𝜒))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 ↔ wb 206 ∧ wa 395 ∨ wo 848 if-wif 1063 ⊼ wnan 1491 ⊽ wnor 1528 ⊤wtru 1541 caddwcad 1606 | 
| 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 207 df-an 396 df-or 849 df-ifp 1064 df-3or 1088 df-3an 1089 df-nan 1492 df-xor 1512 df-nor 1529 df-tru 1543 df-cad 1607 | 
| This theorem is referenced by: (None) | 
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