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Theorem dedlema 885
Description: Lemma for iftrue 3361. (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 641 . . 3 ((𝜓𝜑) → ((𝜓𝜑) ∨ (𝜒 ∧ ¬ 𝜑)))
21expcom 113 . 2 (𝜑 → (𝜓 → ((𝜓𝜑) ∨ (𝜒 ∧ ¬ 𝜑))))
3 simpl 106 . . . 4 ((𝜓𝜑) → 𝜓)
43a1i 9 . . 3 (𝜑 → ((𝜓𝜑) → 𝜓))
5 pm2.24 559 . . . 4 (𝜑 → (¬ 𝜑𝜓))
65adantld 267 . . 3 (𝜑 → ((𝜒 ∧ ¬ 𝜑) → 𝜓))
74, 6jaod 645 . 2 (𝜑 → (((𝜓𝜑) ∨ (𝜒 ∧ ¬ 𝜑)) → 𝜓))
82, 7impbid 124 1 (𝜑 → (𝜓 ↔ ((𝜓𝜑) ∨ (𝜒 ∧ ¬ 𝜑))))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 101  wb 102  wo 637
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in2 553  ax-io 638
This theorem depends on definitions:  df-bi 114
This theorem is referenced by:  iftrue  3361
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