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Definition df-nan 1491
Description: Define incompatibility, or alternative denial ("not-and" or "nand"). See dfnan2 1493 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 398), (df-or 847), (wi 4), and (df-xor 1511). (Contributed by Jeff Hoffman, 19-Nov-2007.)
Assertion
Ref Expression
df-nan ((𝜑𝜓) ↔ ¬ (𝜑𝜓))

Detailed syntax breakdown of Definition df-nan
StepHypRef Expression
1 wph . . 3 wff 𝜑
2 wps . . 3 wff 𝜓
31, 2wnan 1490 . 2 wff (𝜑𝜓)
41, 2wa 397 . . 3 wff (𝜑𝜓)
54wn 3 . 2 wff ¬ (𝜑𝜓)
63, 5wb 205 1 wff ((𝜑𝜓) ↔ ¬ (𝜑𝜓))
Colors of variables: wff setvar class
This definition is referenced by:  nanan  1492  dfnan2  1493  nanor  1494  nanbi  1499  xornan2  1520  trunanfal  1584  nic-mpALT  1675  nic-ax  1676  nic-axALT  1677  nfnan  1904  elnanel  9544  naim1  34864  naim2  34865  df3nandALT1  34874  imnand2  34877  waj-ax  34889  lukshef-ax2  34890  arg-ax  34891  nandsym1  34897  tsna1  36606  tsna2  36607  tsna3  36608  ifpdfnan  41765  ifpnannanb  41786  nanorxor  42592  undisjrab  42593
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