New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  imnan GIF version

Theorem imnan 411
 Description: Express implication in terms of conjunction. (Contributed by NM, 9-Apr-1994.)
Assertion
Ref Expression
imnan ((φ → ¬ ψ) ↔ ¬ (φ ψ))

Proof of Theorem imnan
StepHypRef Expression
1 df-an 360 . 2 ((φ ψ) ↔ ¬ (φ → ¬ ψ))
21con2bii 322 1 ((φ → ¬ ψ) ↔ ¬ (φ ψ))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∧ wa 358 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8 This theorem depends on definitions:  df-bi 177  df-an 360 This theorem is referenced by:  imnani  412  iman  413  ianor  474  nan  563  pm5.17  858  pm5.16  860  dn1  932  nic-ax  1438  nic-axALT  1439  alinexa  1578  dfsb3  2056  ralinexa  2659  pssn2lp  3370  minel  3606  disjsn  3786  ltfinirr  4457  tfinltfin  4501  evenodddisj  4516  funun  5146  imadif  5171  nmembers1lem2  6269  nmembers1lem3  6270
 Copyright terms: Public domain W3C validator