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- ( 𝜓 , 𝜒 , ⊥ ) ↔ ( 𝜓 ∧ 𝜒 ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfifp7 | ⊢ ( if- ( 𝜓 , 𝜒 , ⊥ ) ↔ ( ( ⊥ → 𝜓 ) → ( 𝜓 ∧ 𝜒 ) ) ) | |
2 | falim | ⊢ ( ⊥ → 𝜓 ) | |
3 | 2 | a1bi | ⊢ ( ( 𝜓 ∧ 𝜒 ) ↔ ( ( ⊥ → 𝜓 ) → ( 𝜓 ∧ 𝜒 ) ) ) |
4 | 1 3 | bitr4i | ⊢ ( if- ( 𝜓 , 𝜒 , ⊥ ) ↔ ( 𝜓 ∧ 𝜒 ) ) |