Definition df-nan 1522

Description: Define incompatibility, or alternative denial ("not-and" or "nand"). See dfnan2 1524 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 1573) and the constant false ⊥ (df-fal 1583), we will be able to prove these truth table values: ((⊤ ⊼ ⊤) ↔ ⊥) (trunantru 1611), ((⊤ ⊼ ⊥) ↔ ⊤) (trunanfal 1612), ((⊥ ⊼ ⊤) ↔ ⊤) (falnantru 1613), and ((⊥ ⊼ ⊥) ↔ ⊤) (falnanfal 1614). Contrast with ∧ (df-an 402), ∨ (df-or 862), → (wi 4), and ⊻ (df-xor 1542). (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 1521	. 2 wff (𝜑 ⊼ 𝜓)
4	1, 2	wa 401	. . 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:  nanan  1523  dfnan2  1524  nanor  1525  nanbi  1530  xornan2  1550  trunanfal  1612  nic-mpALT  1705  nic-ax  1706  nic-axALT  1707  nfnan  1933  elnanel  9608  naim1  37177  naim2  37178  df3nandALT1  37187  imnand2  37190  waj-ax  37202  lukshef-ax2  37203  arg-ax  37204  nandsym1  37210  tsna1  39076  tsna2  39077  tsna3  39078  ifpdfnan  44486  ifpnannanb  44507  nanorxor  45288  undisjrab  45289