Definition df-nan 1483

Description: Define incompatibility, or alternative denial ("not-and" or "nand"). 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 1541) and the constant false ⊥ (df-fal 1551), we will be able to prove these truth table values: ((⊤ ⊼ ⊤) ↔ ⊥) (trunantru 1579), ((⊤ ⊼ ⊥) ↔ ⊤) (trunanfal 1580), ((⊥ ⊼ ⊤) ↔ ⊤) (falnantru 1581), and ((⊥ ⊼ ⊥) ↔ ⊤) (falnanfal 1582). Contrast with ∧ (df-an 400), ∨ (df-or 845), → (wi 4), and ⊻ (df-xor 1503). (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 1482	. 2 wff (𝜑 ⊼ 𝜓)
4	1, 2	wa 399	. . 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:  nanan  1484  nanimn  1485  nanor  1486  nanbi  1491  xornan2  1512  trunanfal  1580  nic-mpALT  1674  nic-ax  1675  nic-axALT  1676  nfnan  1901  elnanel  9054  naim1  33850  naim2  33851  df3nandALT1  33860  imnand2  33863  waj-ax  33875  lukshef-ax2  33876  arg-ax  33877  nandsym1  33883  tsna1  35582  tsna2  35583  tsna3  35584  ifpdfnan  40194  ifpnannanb  40215  nanorxor  41009  undisjrab  41010