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Theorem dedlemb 1070
Description: Lemma for weak deduction theorem. See also ifpfal 1098. (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 895 . . 3 ((𝜒 ∧ ¬ 𝜑) → ((𝜓𝜑) ∨ (𝜒 ∧ ¬ 𝜑)))
21expcom 403 . 2 𝜑 → (𝜒 → ((𝜓𝜑) ∨ (𝜒 ∧ ¬ 𝜑))))
3 pm2.21 121 . . . 4 𝜑 → (𝜑𝜒))
43adantld 485 . . 3 𝜑 → ((𝜓𝜑) → 𝜒))
5 simpl 475 . . . 4 ((𝜒 ∧ ¬ 𝜑) → 𝜒)
65a1i 11 . . 3 𝜑 → ((𝜒 ∧ ¬ 𝜑) → 𝜒))
74, 6jaod 886 . 2 𝜑 → (((𝜓𝜑) ∨ (𝜒 ∧ ¬ 𝜑)) → 𝜒))
82, 7impbid 204 1 𝜑 → (𝜒 ↔ ((𝜓𝜑) ∨ (𝜒 ∧ ¬ 𝜑))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wa 385  wo 874
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 199  df-an 386  df-or 875
This theorem is referenced by:  cases2  1071  pm4.42  1077  iffalse  4286
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