Theorem wl-df3maxtru1 37854

Description: Assuming "(n+1)-maxtru1" ↔ ¬ "(n+1)-mintru-2", we can deduce from the recursion formula given in wl-df-3mintru2 37846, 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), (𝜓 ⊼ 𝜒)))

Proof of Theorem wl-df3maxtru1

Step	Hyp	Ref	Expression
1		cadnot 1622	. 2 ⊢ (¬ cadd(𝜑, 𝜓, 𝜒) ↔ cadd(¬ 𝜑, ¬ 𝜓, ¬ 𝜒))
2		wl-df-3mintru2 37846	. 2 ⊢ (cadd(¬ 𝜑, ¬ 𝜓, ¬ 𝜒) ↔ if-(¬ 𝜑, (¬ 𝜓 ∨ ¬ 𝜒), (¬ 𝜓 ∧ ¬ 𝜒)))
3		ifpn 1079	. . 3 ⊢ (if-(𝜑, (𝜓 ⊽ 𝜒), (𝜓 ⊼ 𝜒)) ↔ if-(¬ 𝜑, (𝜓 ⊼ 𝜒), (𝜓 ⊽ 𝜒)))
4		nanor 1502	. . . . . 6 ⊢ ((𝜓 ⊼ 𝜒) ↔ (¬ 𝜓 ∨ ¬ 𝜒))
5	4	a1i 11	. . . . 5 ⊢ (⊤ → ((𝜓 ⊼ 𝜒) ↔ (¬ 𝜓 ∨ ¬ 𝜒)))
6		df-nor 1536	. . . . . . 7 ⊢ ((𝜓 ⊽ 𝜒) ↔ ¬ (𝜓 ∨ 𝜒))
7		ioran 991	. . . . . . 7 ⊢ (¬ (𝜓 ∨ 𝜒) ↔ (¬ 𝜓 ∧ ¬ 𝜒))
8	6, 7	bitri 276	. . . . . 6 ⊢ ((𝜓 ⊽ 𝜒) ↔ (¬ 𝜓 ∧ ¬ 𝜒))
9	8	a1i 11	. . . . 5 ⊢ (⊤ → ((𝜓 ⊽ 𝜒) ↔ (¬ 𝜓 ∧ ¬ 𝜒)))
10	5, 9	ifpbi23d 1085	. . . 4 ⊢ (⊤ → (if-(¬ 𝜑, (𝜓 ⊼ 𝜒), (𝜓 ⊽ 𝜒)) ↔ if-(¬ 𝜑, (¬ 𝜓 ∨ ¬ 𝜒), (¬ 𝜓 ∧ ¬ 𝜒))))
11	10	mptru 1554	. . 3 ⊢ (if-(¬ 𝜑, (𝜓 ⊼ 𝜒), (𝜓 ⊽ 𝜒)) ↔ if-(¬ 𝜑, (¬ 𝜓 ∨ ¬ 𝜒), (¬ 𝜓 ∧ ¬ 𝜒)))
12	3, 11	bitr2i 277	. 2 ⊢ (if-(¬ 𝜑, (¬ 𝜓 ∨ ¬ 𝜒), (¬ 𝜓 ∧ ¬ 𝜒)) ↔ if-(𝜑, (𝜓 ⊽ 𝜒), (𝜓 ⊼ 𝜒)))
13	1, 2, 12	3bitri 298	1 ⊢ (¬ cadd(𝜑, 𝜓, 𝜒) ↔ if-(𝜑, (𝜓 ⊽ 𝜒), (𝜓 ⊼ 𝜒)))

Syntax hints:  ¬ wn 3   ↔ wb 207   ∧ wa 396   ∨ wo 853  if-wif 1068   ⊼ wnan 1498   ⊽ wnor 1535  ⊤wtru 1548  caddwcad 1613

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 208  df-an 397  df-or 854  df-ifp 1069  df-3or 1093  df-3an 1094  df-nan 1499  df-xor 1519  df-nor 1536  df-tru 1550  df-cad 1614

This theorem is referenced by: (None)