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Theorem wl-df3maxtru1 37493
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.)

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

Proof of Theorem wl-df3maxtru1
StepHypRef 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), (𝜓𝜒)) ↔ if-(¬ 𝜑, (𝜓𝜒), (𝜓 𝜒)))
4 nanor 1495 . . . . . 6 ((𝜓𝜒) ↔ (¬ 𝜓 ∨ ¬ 𝜒))
54a1i 11 . . . . 5 (⊤ → ((𝜓𝜒) ↔ (¬ 𝜓 ∨ ¬ 𝜒)))
6 df-nor 1529 . . . . . . 7 ((𝜓 𝜒) ↔ ¬ (𝜓𝜒))
7 ioran 986 . . . . . . 7 (¬ (𝜓𝜒) ↔ (¬ 𝜓 ∧ ¬ 𝜒))
86, 7bitri 275 . . . . . 6 ((𝜓 𝜒) ↔ (¬ 𝜓 ∧ ¬ 𝜒))
98a1i 11 . . . . 5 (⊤ → ((𝜓 𝜒) ↔ (¬ 𝜓 ∧ ¬ 𝜒)))
105, 9ifpbi23d 1080 . . . 4 (⊤ → (if-(¬ 𝜑, (𝜓𝜒), (𝜓 𝜒)) ↔ if-(¬ 𝜑, (¬ 𝜓 ∨ ¬ 𝜒), (¬ 𝜓 ∧ ¬ 𝜒))))
1110mptru 1547 . . 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 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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