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Theorem dedlemb 912
Description: Lemma for iffalse 3377. (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 665 . . 3 ((𝜒 ∧ ¬ 𝜑) → ((𝜓𝜑) ∨ (𝜒 ∧ ¬ 𝜑)))
21expcom 114 . 2 𝜑 → (𝜒 → ((𝜓𝜑) ∨ (𝜒 ∧ ¬ 𝜑))))
3 pm2.21 580 . . . 4 𝜑 → (𝜑𝜒))
43adantld 272 . . 3 𝜑 → ((𝜓𝜑) → 𝜒))
5 simpl 107 . . . 4 ((𝜒 ∧ ¬ 𝜑) → 𝜒)
65a1i 9 . . 3 𝜑 → ((𝜒 ∧ ¬ 𝜑) → 𝜒))
74, 6jaod 670 . 2 𝜑 → (((𝜓𝜑) ∨ (𝜒 ∧ ¬ 𝜑)) → 𝜒))
82, 7impbid 127 1 𝜑 → (𝜒 ↔ ((𝜓𝜑) ∨ (𝜒 ∧ ¬ 𝜑))))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  wb 103  wo 662
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in2 578  ax-io 663
This theorem depends on definitions:  df-bi 115
This theorem is referenced by:  iffalse  3377
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