Theorem iman 413

Description: Express implication in terms of conjunction. Theorem 3.4(27) of [Stoll] p. 176. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 30-Oct-2012.)

Assertion

Ref	Expression
iman	⊢ ((φ → ψ) ↔ ¬ (φ ∧ ¬ ψ))

Proof of Theorem iman

Step	Hyp	Ref	Expression
1		notnot 282	. . 3 ⊢ (ψ ↔ ¬ ¬ ψ)
2	1	imbi2i 303	. 2 ⊢ ((φ → ψ) ↔ (φ → ¬ ¬ ψ))
3		imnan 411	. 2 ⊢ ((φ → ¬ ¬ ψ) ↔ ¬ (φ ∧ ¬ ψ))
4	2, 3	bitri 240	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:  annim  414  pm3.24  852  xor  861  nannan  1291  nic-mpALT  1437  nic-axALT  1439  difdif  3393  dfss4  3490  difin  3493  npss0  3590  ssdif0  3610  difin0ss  3617  inssdif0  3618  dfif2  3665  dfimak2  4299