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Theorem wl-df3maxtru1 37475
Description: Assuming "(n+1)-maxtru1" ↔ ¬ "(n+1)-mintru-2", we can deduce from the recursion formula given in wl-df-3mintru2 37467, 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.)

Assertion
Ref Expression
wl-df3maxtru1 (¬ cadd(𝜑, 𝜓, 𝜒) ↔ if-(𝜑, (𝜓 𝜒), (𝜓𝜒)))

Proof of Theorem wl-df3maxtru1
StepHypRef Expression
1 cadnot 1612 . 2 (¬ cadd(𝜑, 𝜓, 𝜒) ↔ cadd(¬ 𝜑, ¬ 𝜓, ¬ 𝜒))
2 wl-df-3mintru2 37467 . 2 (cadd(¬ 𝜑, ¬ 𝜓, ¬ 𝜒) ↔ if-(¬ 𝜑, (¬ 𝜓 ∨ ¬ 𝜒), (¬ 𝜓 ∧ ¬ 𝜒)))
3 ifpn 1073 . . 3 (if-(𝜑, (𝜓 𝜒), (𝜓𝜒)) ↔ if-(¬ 𝜑, (𝜓𝜒), (𝜓 𝜒)))
4 nanor 1492 . . . . . 6 ((𝜓𝜒) ↔ (¬ 𝜓 ∨ ¬ 𝜒))
54a1i 11 . . . . 5 (⊤ → ((𝜓𝜒) ↔ (¬ 𝜓 ∨ ¬ 𝜒)))
6 df-nor 1526 . . . . . . 7 ((𝜓 𝜒) ↔ ¬ (𝜓𝜒))
7 ioran 985 . . . . . . 7 (¬ (𝜓𝜒) ↔ (¬ 𝜓 ∧ ¬ 𝜒))
86, 7bitri 275 . . . . . 6 ((𝜓 𝜒) ↔ (¬ 𝜓 ∧ ¬ 𝜒))
98a1i 11 . . . . 5 (⊤ → ((𝜓 𝜒) ↔ (¬ 𝜓 ∧ ¬ 𝜒)))
105, 9ifpbi23d 1079 . . . 4 (⊤ → (if-(¬ 𝜑, (𝜓𝜒), (𝜓 𝜒)) ↔ if-(¬ 𝜑, (¬ 𝜓 ∨ ¬ 𝜒), (¬ 𝜓 ∧ ¬ 𝜒))))
1110mptru 1544 . . 3 (if-(¬ 𝜑, (𝜓𝜒), (𝜓 𝜒)) ↔ if-(¬ 𝜑, (¬ 𝜓 ∨ ¬ 𝜒), (¬ 𝜓 ∧ ¬ 𝜒)))
123, 11bitr2i 276 . 2 (if-(¬ 𝜑, (¬ 𝜓 ∨ ¬ 𝜒), (¬ 𝜓 ∧ ¬ 𝜒)) ↔ if-(𝜑, (𝜓 𝜒), (𝜓𝜒)))
131, 2, 123bitri 297 1 (¬ cadd(𝜑, 𝜓, 𝜒) ↔ if-(𝜑, (𝜓 𝜒), (𝜓𝜒)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 206  wa 395  wo 847  if-wif 1062  wnan 1488   wnor 1525  wtru 1538  caddwcad 1603
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 848  df-ifp 1063  df-3or 1087  df-3an 1088  df-nan 1489  df-xor 1509  df-nor 1526  df-tru 1540  df-cad 1604
This theorem is referenced by: (None)
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