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Mathbox for Wolf Lammen |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > wl-2mintru1 | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
wl-2mintru1 | ⊢ (if-(𝜓, ⊤, 𝜒) ↔ (𝜓 ∨ 𝜒)) |
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)) |
Colors of variables: wff setvar class |
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) |
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