Definition df-nor 1559

Description: Define joint denial ("not-or" or "nor"). After we define the constant true ⊤ (df-tru 1573) and the constant false ⊥ (df-fal 1583), we will be able to prove these truth table values: ((⊤ ⊽ ⊤) ↔ ⊥) (trunortru 1619), ((⊤ ⊽ ⊥) ↔ ⊥) (trunorfal 1620), ((⊥ ⊽ ⊤) ↔ ⊥) (falnortru 1621), and ((⊥ ⊽ ⊥) ↔ ⊤) (falnorfal 1622). Contrast with ∧ (df-an 402), ∨ (df-or 862), → (wi 4), ⊼ (df-nan 1522), and ⊻ (df-xor 1542). (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 1558	. 2 wff (𝜑 ⊽ 𝜓)
4	1, 2	wo 861	. . 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 used by:  norcom  1560  nornot  1561  noran  1562  noror  1563  norasslem1  1564  norass  1567  trunortru  1619  trunorfal  1620  falnorfal  1622  wl-df3maxtru1  38233