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

Theorem dedlema 954
 Description: Lemma for iftrue 3484. (Contributed by NM, 26-Jun-2002.) (Proof shortened by Andrew Salmon, 7-May-2011.)
Assertion
Ref Expression
dedlema (𝜑 → (𝜓 ↔ ((𝜓𝜑) ∨ (𝜒 ∧ ¬ 𝜑))))

Proof of Theorem dedlema
StepHypRef Expression
1 orc 702 . . 3 ((𝜓𝜑) → ((𝜓𝜑) ∨ (𝜒 ∧ ¬ 𝜑)))
21expcom 115 . 2 (𝜑 → (𝜓 → ((𝜓𝜑) ∨ (𝜒 ∧ ¬ 𝜑))))
3 simpl 108 . . . 4 ((𝜓𝜑) → 𝜓)
43a1i 9 . . 3 (𝜑 → ((𝜓𝜑) → 𝜓))
5 pm2.24 611 . . . 4 (𝜑 → (¬ 𝜑𝜓))
65adantld 276 . . 3 (𝜑 → ((𝜒 ∧ ¬ 𝜑) → 𝜓))
74, 6jaod 707 . 2 (𝜑 → (((𝜓𝜑) ∨ (𝜒 ∧ ¬ 𝜑)) → 𝜓))
82, 7impbid 128 1 (𝜑 → (𝜓 ↔ ((𝜓𝜑) ∨ (𝜒 ∧ ¬ 𝜑))))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104   ∨ wo 698 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-in2 605  ax-io 699 This theorem depends on definitions:  df-bi 116 This theorem is referenced by:  iftrue  3484
 Copyright terms: Public domain W3C validator