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Theorem dedlema 1030
 Description: Lemma for weak deduction theorem. See also ifptru 1059. (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 847 . . 3 ((𝜓𝜑) → ((𝜓𝜑) ∨ (𝜒 ∧ ¬ 𝜑)))
21expcom 398 . 2 (𝜑 → (𝜓 → ((𝜓𝜑) ∨ (𝜒 ∧ ¬ 𝜑))))
3 simpl 468 . . . 4 ((𝜓𝜑) → 𝜓)
43a1i 11 . . 3 (𝜑 → ((𝜓𝜑) → 𝜓))
5 pm2.24 122 . . . 4 (𝜑 → (¬ 𝜑𝜓))
65adantld 474 . . 3 (𝜑 → ((𝜒 ∧ ¬ 𝜑) → 𝜓))
74, 6jaod 839 . 2 (𝜑 → (((𝜓𝜑) ∨ (𝜒 ∧ ¬ 𝜑)) → 𝜓))
82, 7impbid 202 1 (𝜑 → (𝜓 ↔ ((𝜓𝜑) ∨ (𝜒 ∧ ¬ 𝜑))))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 382   ∨ wo 826 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827 This theorem is referenced by:  cases2  1032  pm4.42  1040
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