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

Theorem oibabs 800
 Description: Absorption of disjunction into equivalence. (Contributed by NM, 6-Aug-1995.) (Proof shortened by Wolf Lammen, 3-Nov-2013.)
Assertion
Ref Expression
oibabs (((𝜑𝜓) → (𝜑𝜓)) ↔ (𝜑𝜓))

Proof of Theorem oibabs
StepHypRef Expression
1 pm2.67-2 634 . . . 4 (((𝜑𝜓) → (𝜑𝜓)) → (𝜑 → (𝜑𝜓)))
21ibd 167 . . 3 (((𝜑𝜓) → (𝜑𝜓)) → (𝜑𝜓))
3 olc 632 . . . . 5 (𝜓 → (𝜑𝜓))
43imim1i 54 . . . 4 (((𝜑𝜓) → (𝜑𝜓)) → (𝜓 → (𝜑𝜓)))
5 ibibr 235 . . . 4 ((𝜓𝜑) ↔ (𝜓 → (𝜑𝜓)))
64, 5sylibr 137 . . 3 (((𝜑𝜓) → (𝜑𝜓)) → (𝜓𝜑))
72, 6impbid 120 . 2 (((𝜑𝜓) → (𝜑𝜓)) → (𝜑𝜓))
8 ax-1 5 . 2 ((𝜑𝜓) → ((𝜑𝜓) → (𝜑𝜓)))
97, 8impbii 117 1 (((𝜑𝜓) → (𝜑𝜓)) ↔ (𝜑𝜓))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98   ∨ wo 629 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630 This theorem depends on definitions:  df-bi 110 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator