Theorem wl-df3maxtru1 36373

Description: Assuming "(n+1)-maxtru1" ↔ ¬ "(n+1)-mintru-2", we can deduce from the recursion formula given in wl-df-3mintru2 36365, 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 simultaniously 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 1617	. 2 ⊢ (¬ cadd(𝜑, 𝜓, 𝜒) ↔ cadd(¬ 𝜑, ¬ 𝜓, ¬ 𝜒))
2		wl-df-3mintru2 36365	. 2 ⊢ (cadd(¬ 𝜑, ¬ 𝜓, ¬ 𝜒) ↔ if-(¬ 𝜑, (¬ 𝜓 ∨ ¬ 𝜒), (¬ 𝜓 ∧ ¬ 𝜒)))
3		ifpn 1073	. . 3 ⊢ (if-(𝜑, (𝜓 ⊽ 𝜒), (𝜓 ⊼ 𝜒)) ↔ if-(¬ 𝜑, (𝜓 ⊼ 𝜒), (𝜓 ⊽ 𝜒)))
4		nanor 1494	. . . . . 6 ⊢ ((𝜓 ⊼ 𝜒) ↔ (¬ 𝜓 ∨ ¬ 𝜒))
5	4	a1i 11	. . . . 5 ⊢ (⊤ → ((𝜓 ⊼ 𝜒) ↔ (¬ 𝜓 ∨ ¬ 𝜒)))
6		df-nor 1530	. . . . . . 7 ⊢ ((𝜓 ⊽ 𝜒) ↔ ¬ (𝜓 ∨ 𝜒))
7		ioran 983	. . . . . . 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 1081	. . . 4 ⊢ (⊤ → (if-(¬ 𝜑, (𝜓 ⊼ 𝜒), (𝜓 ⊽ 𝜒)) ↔ if-(¬ 𝜑, (¬ 𝜓 ∨ ¬ 𝜒), (¬ 𝜓 ∧ ¬ 𝜒))))
11	10	mptru 1549	. . 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)))

Syntax hints:  ¬ wn 3   ↔ wb 205   ∧ wa 397   ∨ wo 846  if-wif 1062   ⊼ wnan 1490   ⊽ wnor 1529  ⊤wtru 1543  caddwcad 1608

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 206  df-an 398  df-or 847  df-ifp 1063  df-3or 1089  df-3an 1090  df-nan 1491  df-xor 1511  df-nor 1530  df-tru 1545  df-cad 1609

This theorem is referenced by: (None)