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Mirrors > Home > MPE Home > Th. List > df-nor | Structured version Visualization version GIF version |
Description: Define joint denial ("not-or" or "nor"). After we define the constant true ⊤ (df-tru 1544) and the constant false ⊥ (df-fal 1554), we will be able to prove these truth table values: ((⊤ ⊽ ⊤) ↔ ⊥) (trunortru 1590), ((⊤ ⊽ ⊥) ↔ ⊥) (trunorfal 1591), ((⊥ ⊽ ⊤) ↔ ⊥) (falnortru 1592), and ((⊥ ⊽ ⊥) ↔ ⊤) (falnorfal 1593). Contrast with ∧ (df-an 397), ∨ (df-or 846), → (wi 4), ⊼ (df-nan 1490), and ⊻ (df-xor 1510). (Contributed by Remi, 25-Oct-2023.) |
Ref | Expression |
---|---|
df-nor | ⊢ ((𝜑 ⊽ 𝜓) ↔ ¬ (𝜑 ∨ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wph | . . 3 wff 𝜑 | |
2 | wps | . . 3 wff 𝜓 | |
3 | 1, 2 | wnor 1528 | . 2 wff (𝜑 ⊽ 𝜓) |
4 | 1, 2 | wo 845 | . . 3 wff (𝜑 ∨ 𝜓) |
5 | 4 | wn 3 | . 2 wff ¬ (𝜑 ∨ 𝜓) |
6 | 3, 5 | wb 205 | 1 wff ((𝜑 ⊽ 𝜓) ↔ ¬ (𝜑 ∨ 𝜓)) |
Colors of variables: wff setvar class |
This definition is referenced by: norcom 1530 norcomOLD 1531 nornot 1532 noran 1533 noror 1534 norasslem1 1535 norass 1538 trunortru 1590 trunorfal 1591 falnorfal 1593 wl-df3maxtru1 36368 |
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