![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 1542) and the constant false ⊥ (df-fal 1552), we will be able to prove these truth table values: ((⊤ ⊽ ⊤) ↔ ⊥) (trunortru 1588), ((⊤ ⊽ ⊥) ↔ ⊥) (trunorfal 1589), ((⊥ ⊽ ⊤) ↔ ⊥) (falnortru 1590), and ((⊥ ⊽ ⊥) ↔ ⊤) (falnorfal 1591). Contrast with ∧ (df-an 395), ∨ (df-or 844), → (wi 4), ⊼ (df-nan 1488), and ⊻ (df-xor 1508). (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 1526 | . 2 wff (𝜑 ⊽ 𝜓) |
4 | 1, 2 | wo 843 | . . 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 1528 norcomOLD 1529 nornot 1530 noran 1531 noror 1532 norasslem1 1533 norass 1536 trunortru 1588 trunorfal 1589 falnorfal 1591 wl-df3maxtru1 36678 |
Copyright terms: Public domain | W3C validator |