Definition df-nan 1494

Description: Define incompatibility, or alternative denial ("not-and" or "nand"). See dfnan2 1496 for an alternative. This is also called the Sheffer stroke, represented by a vertical bar, but we use a different symbol to avoid ambiguity with other uses of the vertical bar. In the second edition of Principia Mathematica (1927), Russell and Whitehead used the Sheffer stroke and suggested it as a replacement for the "or" and "not" operations of the first edition. However, in practice, "or" and "not" are more widely used. After we define the constant true ⊤ (df-tru 1545) and the constant false ⊥ (df-fal 1555), we will be able to prove these truth table values: ((⊤ ⊼ ⊤) ↔ ⊥) (trunantru 1583), ((⊤ ⊼ ⊥) ↔ ⊤) (trunanfal 1584), ((⊥ ⊼ ⊤) ↔ ⊤) (falnantru 1585), and ((⊥ ⊼ ⊥) ↔ ⊤) (falnanfal 1586). Contrast with ∧ (df-an 396), ∨ (df-or 849), → (wi 4), and ⊻ (df-xor 1514). (Contributed by Jeff Hoffman, 19-Nov-2007.)

Assertion

Ref	Expression
df-nan	⊢ ((𝜑 ⊼ 𝜓) ↔ ¬ (𝜑 ∧ 𝜓))

Detailed syntax breakdown of Definition df-nan

Step	Hyp	Ref	Expression
1		wph	. . 3 wff 𝜑
2		wps	. . 3 wff 𝜓
3	1, 2	wnan 1493	. 2 wff (𝜑 ⊼ 𝜓)
4	1, 2	wa 395	. . 3 wff (𝜑 ∧ 𝜓)
5	4	wn 3	. 2 wff ¬ (𝜑 ∧ 𝜓)
6	3, 5	wb 206	1 wff ((𝜑 ⊼ 𝜓) ↔ ¬ (𝜑 ∧ 𝜓))

This definition is referenced by:  nanan  1495  dfnan2  1496  nanor  1497  nanbi  1502  xornan2  1522  trunanfal  1584  nic-mpALT  1674  nic-ax  1675  nic-axALT  1676  nfnan  1902  elnanel  9528  naim1  36605  naim2  36606  df3nandALT1  36615  imnand2  36618  waj-ax  36630  lukshef-ax2  36631  arg-ax  36632  nandsym1  36638  tsna1  38395  tsna2  38396  tsna3  38397  ifpdfnan  43842  ifpnannanb  43863  nanorxor  44661  undisjrab  44662