Definition df-nor 1556

Description: Define joint denial ("not-or" or "nor"). After we define the constant true ⊤ (df-tru 1570) and the constant false ⊥ (df-fal 1580), we will be able to prove these truth table values: ((⊤ ⊽ ⊤) ↔ ⊥) (trunortru 1616), ((⊤ ⊽ ⊥) ↔ ⊥) (trunorfal 1617), ((⊥ ⊽ ⊤) ↔ ⊥) (falnortru 1618), and ((⊥ ⊽ ⊥) ↔ ⊤) (falnorfal 1619). Contrast with ∧ (df-an 401), ∨ (df-or 861), → (wi 4), ⊼ (df-nan 1519), and ⊻ (df-xor 1539). (Contributed by Remi, 25-Oct-2023.)

Assertion

Ref	Expression
df-nor	⊢ ((𝜑 ⊽ 𝜓) ↔ ¬ (𝜑 ∨ 𝜓))

Detailed syntax breakdown of Definition df-nor

Step	Hyp	Ref	Expression
1		wph	. . 3 wff 𝜑
2		wps	. . 3 wff 𝜓
3	1, 2	wnor 1555	. 2 wff (𝜑 ⊽ 𝜓)
4	1, 2	wo 860	. . 3 wff (𝜑 ∨ 𝜓)
5	4	wn 3	. 2 wff ¬ (𝜑 ∨ 𝜓)
6	3, 5	wb 209	1 wff ((𝜑 ⊽ 𝜓) ↔ ¬ (𝜑 ∨ 𝜓))

This definition is referenced by:  norcom  1557  nornot  1558  noran  1559  noror  1560  norasslem1  1561  norass  1564  trunortru  1616  trunorfal  1617  falnorfal  1619  wl-df3maxtru1  38026