Theorem wl-2mintru1

Description: Using the recursion formula

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

for "2-mintru-1" (meaning "at least 1 out of 2 inputs is true") by plugging in n = 1, m = 0, and simplifying. The expression "1-mintru-0" is a base case (meaning at least zero inputs out of 1 are true), evaluating to T. , and wl-1mintru1 shows "1-mintru-1" is equivalent to the only input.

Negating an "n-mintru1" operation means: All n inputs ph .. th are false. This is also conveniently expressed as -. ( ph \/ .. \/ th ) , in accordance with the result here. (Contributed by Wolf Lammen, 10-May-2024)

	Ref	Expression
Assertion	wl-2mintru1	⊢ ( if- ( 𝜓 , ⊤ , 𝜒 ) ↔ ( 𝜓 ∨ 𝜒 ) )

Proof

Step	Hyp	Ref	Expression
1		dfifp3	⊢ ( if- ( 𝜓 , ⊤ , 𝜒 ) ↔ ( ( 𝜓 → ⊤ ) ∧ ( 𝜓 ∨ 𝜒 ) ) )
2		trud	⊢ ( 𝜓 → ⊤ )
3	2	bitru	⊢ ( ( 𝜓 → ⊤ ) ↔ ⊤ )
4	3	anbi1i	⊢ ( ( ( 𝜓 → ⊤ ) ∧ ( 𝜓 ∨ 𝜒 ) ) ↔ ( ⊤ ∧ ( 𝜓 ∨ 𝜒 ) ) )
5		truan	⊢ ( ( ⊤ ∧ ( 𝜓 ∨ 𝜒 ) ) ↔ ( 𝜓 ∨ 𝜒 ) )
6	1 4 5	3bitri	⊢ ( if- ( 𝜓 , ⊤ , 𝜒 ) ↔ ( 𝜓 ∨ 𝜒 ) )