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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 ((φψ) ↔ ¬ (φ ¬ ψ))

Proof of Theorem iman
StepHypRef Expression
1 notnot 282 . . 3 (ψ ↔ ¬ ¬ ψ)
21imbi2i 303 . 2 ((φψ) ↔ (φ → ¬ ¬ ψ))
3 imnan 411 . 2 ((φ → ¬ ¬ ψ) ↔ ¬ (φ ¬ ψ))
42, 3bitri 240 1 ((φψ) ↔ ¬ (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-1 5  ax-2 6  ax-3 7  ax-mp 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  3392  dfss4  3489  difin  3492  npss0  3589  ssdif0  3609  difin0ss  3616  inssdif0  3617  dfif2  3664  dfimak2  4298
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