Theorem wl-2mintru2

Description: Using the recursion formula

"(n+1)-mintru-(m+1)" <-> if- ( ph , "n-mintru-m" , "n-mintru-(m+1)" )

for "2-mintru-2" (meaning "2 out of 2 inputs are true") by plugging in n = 1, m = 1, and simplifying. See wl-1mintru1 and wl-1mintru2 to see that "1-mintru-1" / "1-mintru-2" evaluate to ch / F. respectively.

Negating a "n-mintru2" operation means 'at most one input is true', so all inputs exclude each other mutually. Such an exclusion is expressed by a NAND operation ( ph -/\ ps ) , not by a XOR. Applying this idea here (n = 2) yields the expected NAND in case of a pair of inputs. (Contributed by Wolf Lammen, 10-May-2024)

	Ref	Expression
Assertion	wl-2mintru2	$$⊢if-\left(\psi ,\chi ,\perp \right)\leftrightarrow \left(\psi \wedge \chi \right)$$

Proof

Step	Hyp	Ref	Expression
1		dfifp7	$$⊢if-\left(\psi ,\chi ,\perp \right)\leftrightarrow \left(\left(\perp \to \psi \right)\to \left(\psi \wedge \chi \right)\right)$$
2		falim	$$⊢\perp \to \psi$$
3	2	a1bi	$$⊢\left(\psi \wedge \chi \right)\leftrightarrow \left(\left(\perp \to \psi \right)\to \left(\psi \wedge \chi \right)\right)$$
4	1 3	bitr4i	$$⊢if-\left(\psi ,\chi ,\perp \right)\leftrightarrow \left(\psi \wedge \chi \right)$$