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Theorem dedlema 993
 Description: Lemma for weak deduction theorem. See also ifptru 1017. (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 399 . . 3 ((𝜓𝜑) → ((𝜓𝜑) ∨ (𝜒 ∧ ¬ 𝜑)))
21expcom 450 . 2 (𝜑 → (𝜓 → ((𝜓𝜑) ∨ (𝜒 ∧ ¬ 𝜑))))
3 simpl 472 . . . 4 ((𝜓𝜑) → 𝜓)
43a1i 11 . . 3 (𝜑 → ((𝜓𝜑) → 𝜓))
5 pm2.24 120 . . . 4 (𝜑 → (¬ 𝜑𝜓))
65adantld 482 . . 3 (𝜑 → ((𝜒 ∧ ¬ 𝜑) → 𝜓))
74, 6jaod 394 . 2 (𝜑 → (((𝜓𝜑) ∨ (𝜒 ∧ ¬ 𝜑)) → 𝜓))
82, 7impbid 201 1 (𝜑 → (𝜓 ↔ ((𝜓𝜑) ∨ (𝜒 ∧ ¬ 𝜑))))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383 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 196  df-or 384  df-an 385 This theorem is referenced by:  pm4.42  995  elimhOLD  1027  dedtOLD  1028
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