Theorem wl-2mintru1 35172

Description: Using the recursion formula

"(n+1)-mintru-(m+1)" ↔ if-(𝜑, "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 ⊤, and wl-1mintru1 35170 shows "1-mintru-1" is equivalent to the only input.

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

Assertion

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

Proof of Theorem wl-2mintru1

Step	Hyp	Ref	Expression
1		dfifp3 1062	. 2 ⊢ (if-(𝜓, ⊤, 𝜒) ↔ ((𝜓 → ⊤) ∧ (𝜓 ∨ 𝜒)))
2		trud 1549	. . . 4 ⊢ (𝜓 → ⊤)
3	2	bitru 1548	. . 3 ⊢ ((𝜓 → ⊤) ↔ ⊤)
4	3	anbi1i 627	. 2 ⊢ (((𝜓 → ⊤) ∧ (𝜓 ∨ 𝜒)) ↔ (⊤ ∧ (𝜓 ∨ 𝜒)))
5		truan 1550	. 2 ⊢ ((⊤ ∧ (𝜓 ∨ 𝜒)) ↔ (𝜓 ∨ 𝜒))
6	1, 4, 5	3bitri 301	1 ⊢ (if-(𝜓, ⊤, 𝜒) ↔ (𝜓 ∨ 𝜒))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 845  if-wif 1059  ⊤wtru 1540

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 210  df-an 401  df-or 846  df-ifp 1060  df-tru 1542

This theorem is referenced by: (None)