Definition df-nan 1294

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 T. (df-tru 1325) and the constant false F. (df-fal 1326), we will be able to prove these truth table values: ( ( T. -/\ T. ) <-> F. ) (trunantru 1360), ( ( T. -/\ F. ) <-> T. ) (trunanfal 1361), ( ( F. -/\ T. ) <-> T. ) (falnantru 1362), and ( ( F. -/\ F. ) <-> T. ) (falnanfal 1363). Contrast with /\ (df-an 361), \/ (df-or 360), -> (wi 4), and \/_ (df-xor 1311) . (Contributed by Jeff Hoffman, 19-Nov-2007.)

Assertion

Ref	Expression
df-nan	|- ( ( ph -/\ ps ) <-> -. ( ph /\ ps ) )

Detailed syntax breakdown of Definition df-nan

Step	Hyp	Ref	Expression
1		wph	. . 3 wff ph
2		wps	. . 3 wff ps
3	1, 2	wnan 1293	. 2 wff ( ph -/\ ps )
4	1, 2	wa 359	. . 3 wff ( ph /\ ps )
5	4	wn 3	. 2 wff -. ( ph /\ ps )
6	3, 5	wb 177	1 wff ( ( ph -/\ ps ) <-> -. ( ph /\ ps ) )

This definition is referenced by:  nanan  1295  nancom  1296  nannan  1297  nannot  1299  nanbi  1300  nanbi1  1301  trunanfal  1361  nic-mpALT  1443  nic-ax  1444  nic-axALT  1445  nfnan  1843  naim1  26038  naim2  26039  df3nandALT1  26050  imnand2  26053  waj-ax  26068  lukshef-ax2  26069  arg-ax  26070  nandsym1  26076