Theorem wl-1mintru1

Description: Using the recursion formula:

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

for "1-mintru-1" (meaning "at least 1 out of 1 input is true") by plugging in n = 0, m = 0, and simplifying. The expressions "0-mintru-0" and "0-mintru-1" are base cases of the recursion, meaning "in a sequence of zero inputs, at least 0 / 1 input is true", respectively equvalent to T. / F. .

Negating an "n-mintru1" operation means: All n inputs ph .. th are false. This is also conveniently expressed as -. ( ph \/ .. \/ th ) . Applying this idea here (n = 1) yields the obvious result that in an input sequence of size 1 only then all will be false, if its single input is. (Contributed by Wolf Lammen, 10-May-2024)

	Ref	Expression
Assertion	wl-1mintru1	⊢ ( if- ( 𝜒 , ⊤ , ⊥ ) ↔ 𝜒 )

Proof

Step	Hyp	Ref	Expression
1		tbtru	⊢ ( 𝜒 ↔ ( 𝜒 ↔ ⊤ ) )
2	1	biimpi	⊢ ( 𝜒 → ( 𝜒 ↔ ⊤ ) )
3		nbfal	⊢ ( ¬ 𝜒 ↔ ( 𝜒 ↔ ⊥ ) )
4	3	biimpi	⊢ ( ¬ 𝜒 → ( 𝜒 ↔ ⊥ ) )
5	2 4	casesifp	⊢ ( 𝜒 ↔ if- ( 𝜒 , ⊤ , ⊥ ) )
6	5	bicomi	⊢ ( if- ( 𝜒 , ⊤ , ⊥ ) ↔ 𝜒 )