ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  imanst GIF version

Theorem imanst 874
Description: Express implication in terms of conjunction. Theorem 3.4(27) of [Stoll] p. 176. (Contributed by NM, 12-Mar-1993.) (Proof shortened by Wolf Lammen, 30-Oct-2012.)
Assertion
Ref Expression
imanst (STAB 𝜓 → ((𝜑𝜓) ↔ ¬ (𝜑 ∧ ¬ 𝜓)))

Proof of Theorem imanst
StepHypRef Expression
1 notnot 619 . . . 4 (𝜓 → ¬ ¬ 𝜓)
2 df-stab 817 . . . . 5 (STAB 𝜓 ↔ (¬ ¬ 𝜓𝜓))
32biimpi 119 . . . 4 (STAB 𝜓 → (¬ ¬ 𝜓𝜓))
41, 3impbid2 142 . . 3 (STAB 𝜓 → (𝜓 ↔ ¬ ¬ 𝜓))
54imbi2d 229 . 2 (STAB 𝜓 → ((𝜑𝜓) ↔ (𝜑 → ¬ ¬ 𝜓)))
6 imnan 680 . 2 ((𝜑 → ¬ ¬ 𝜓) ↔ ¬ (𝜑 ∧ ¬ 𝜓))
75, 6syl6bb 195 1 (STAB 𝜓 → ((𝜑𝜓) ↔ ¬ (𝜑 ∧ ¬ 𝜓)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wb 104  STAB wstab 816
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605
This theorem depends on definitions:  df-bi 116  df-stab 817
This theorem is referenced by:  imandc  875  dfss4st  3313
  Copyright terms: Public domain W3C validator