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

Step	Hyp	Ref	Expression
1		df-an 360	. 2 ⊢ ((φ ∧ ψ) ↔ ¬ (φ → ¬ ψ))
2	1	con2bii 322	1 ⊢ ((φ → ¬ ψ) ↔ ¬ (φ ∧ ψ))

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  2660  pssn2lp  3371  minel  3607  disjsn  3787  ltfinirr  4458  tfinltfin  4502  evenodddisj  4517  funun  5147  imadif  5172  nmembers1lem2  6270  nmembers1lem3  6271