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Mathbox for Wolf Lammen |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > wl-2mintru2 | 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-2" (meaning "2 out of 2 inputs are true") by plugging in n = 1, m = 1, and simplifying. See wl-1mintru1 35170 and wl-1mintru2 35171 to see that "1-mintru-1" / "1-mintru-2" evaluate to 𝜒 / ⊥ respectively. Negating a "n-mintru2" operation means 'at most one input is true', so all inputs exclude each other mutually. Such an exclusion is expressed by a NAND operation (𝜑 ⊼ 𝜓), not by a XOR. Applying this idea here (n = 2) yields the expected NAND in case of a pair of inputs. (Contributed by Wolf Lammen, 10-May-2024.) |
Ref | Expression |
---|---|
wl-2mintru2 | ⊢ (if-(𝜓, 𝜒, ⊥) ↔ (𝜓 ∧ 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfifp7 1066 | . 2 ⊢ (if-(𝜓, 𝜒, ⊥) ↔ ((⊥ → 𝜓) → (𝜓 ∧ 𝜒))) | |
2 | falim 1556 | . . 3 ⊢ (⊥ → 𝜓) | |
3 | 2 | a1bi 367 | . 2 ⊢ ((𝜓 ∧ 𝜒) ↔ ((⊥ → 𝜓) → (𝜓 ∧ 𝜒))) |
4 | 1, 3 | bitr4i 281 | 1 ⊢ (if-(𝜓, 𝜒, ⊥) ↔ (𝜓 ∧ 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 400 if-wif 1059 ⊥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) |
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