Theorem wl-1mintru1 35170

Description: Using the recursion formula:

"(n+1)-mintru-(m+1)" ↔ if-(𝜑, "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 ⊤ / ⊥.

Negating an "n-mintru1" operation means: All n inputs 𝜑.. 𝜃 are false. This is also conveniently expressed as ¬ (𝜑 ∨.. ∨ 𝜃). 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.)

Assertion

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

Proof of Theorem wl-1mintru1

Step	Hyp	Ref	Expression
1		tbtru 1547	. . . 4 ⊢ (𝜒 ↔ (𝜒 ↔ ⊤))
2	1	biimpi 219	. . 3 ⊢ (𝜒 → (𝜒 ↔ ⊤))
3		nbfal 1554	. . . 4 ⊢ (¬ 𝜒 ↔ (𝜒 ↔ ⊥))
4	3	biimpi 219	. . 3 ⊢ (¬ 𝜒 → (𝜒 ↔ ⊥))
5	2, 4	casesifp 1075	. 2 ⊢ (𝜒 ↔ if-(𝜒, ⊤, ⊥))
6	5	bicomi 227	1 ⊢ (if-(𝜒, ⊤, ⊥) ↔ 𝜒)

Syntax hints:  ¬ wn 3   ↔ wb 209  if-wif 1059  ⊤wtru 1540  ⊥wfal 1551

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  df-fal 1552

This theorem is referenced by: (None)