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Theorem dedlema 876
 Description: Lemma for iftrue 3336. (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 633 . . 3 ((𝜓𝜑) → ((𝜓𝜑) ∨ (𝜒 ∧ ¬ 𝜑)))
21expcom 109 . 2 (𝜑 → (𝜓 → ((𝜓𝜑) ∨ (𝜒 ∧ ¬ 𝜑))))
3 simpl 102 . . . 4 ((𝜓𝜑) → 𝜓)
43a1i 9 . . 3 (𝜑 → ((𝜓𝜑) → 𝜓))
5 pm2.24 551 . . . 4 (𝜑 → (¬ 𝜑𝜓))
65adantld 263 . . 3 (𝜑 → ((𝜒 ∧ ¬ 𝜑) → 𝜓))
74, 6jaod 637 . 2 (𝜑 → (((𝜓𝜑) ∨ (𝜒 ∧ ¬ 𝜑)) → 𝜓))
82, 7impbid 120 1 (𝜑 → (𝜓 ↔ ((𝜓𝜑) ∨ (𝜒 ∧ ¬ 𝜑))))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   ↔ 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-in2 545  ax-io 630 This theorem depends on definitions:  df-bi 110 This theorem is referenced by:  iftrue  3336
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