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Theorem dedlemb 1041
Description: Lemma for weak deduction theorem. See also ifpfal 1069. (Contributed by NM, 15-May-1999.) (Proof shortened by Andrew Salmon, 7-May-2011.)
Assertion
Ref Expression
dedlemb 𝜑 → (𝜒 ↔ ((𝜓𝜑) ∨ (𝜒 ∧ ¬ 𝜑))))

Proof of Theorem dedlemb
StepHypRef Expression
1 olc 864 . . 3 ((𝜒 ∧ ¬ 𝜑) → ((𝜓𝜑) ∨ (𝜒 ∧ ¬ 𝜑)))
21expcom 416 . 2 𝜑 → (𝜒 → ((𝜓𝜑) ∨ (𝜒 ∧ ¬ 𝜑))))
3 pm2.21 123 . . . 4 𝜑 → (𝜑𝜒))
43adantld 493 . . 3 𝜑 → ((𝜓𝜑) → 𝜒))
5 simpl 485 . . . 4 ((𝜒 ∧ ¬ 𝜑) → 𝜒)
65a1i 11 . . 3 𝜑 → ((𝜒 ∧ ¬ 𝜑) → 𝜒))
74, 6jaod 855 . 2 𝜑 → (((𝜓𝜑) ∨ (𝜒 ∧ ¬ 𝜑)) → 𝜒))
82, 7impbid 214 1 𝜑 → (𝜒 ↔ ((𝜓𝜑) ∨ (𝜒 ∧ ¬ 𝜑))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 208  wa 398  wo 843
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 209  df-an 399  df-or 844
This theorem is referenced by:  cases2  1042  pm4.42  1048  iffalse  4475
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