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Theorem ecase13d 32609
Description: Deduction for elimination by cases. (Contributed by Jeff Hankins, 18-Aug-2009.)
Hypotheses
Ref Expression
ecase13d.1 (𝜑 → ¬ 𝜒)
ecase13d.2 (𝜑 → ¬ 𝜃)
ecase13d.3 (𝜑 → (𝜒𝜓𝜃))
Assertion
Ref Expression
ecase13d (𝜑𝜓)

Proof of Theorem ecase13d
StepHypRef Expression
1 ecase13d.2 . 2 (𝜑 → ¬ 𝜃)
2 ecase13d.1 . . . 4 (𝜑 → ¬ 𝜒)
3 ecase13d.3 . . . . 5 (𝜑 → (𝜒𝜓𝜃))
4 3orass 1075 . . . . . 6 ((𝜒𝜓𝜃) ↔ (𝜒 ∨ (𝜓𝜃)))
5 df-or 384 . . . . . 6 ((𝜒 ∨ (𝜓𝜃)) ↔ (¬ 𝜒 → (𝜓𝜃)))
64, 5bitri 264 . . . . 5 ((𝜒𝜓𝜃) ↔ (¬ 𝜒 → (𝜓𝜃)))
73, 6sylib 208 . . . 4 (𝜑 → (¬ 𝜒 → (𝜓𝜃)))
82, 7mpd 15 . . 3 (𝜑 → (𝜓𝜃))
9 orcom 401 . . . 4 ((𝜓𝜃) ↔ (𝜃𝜓))
10 df-or 384 . . . 4 ((𝜃𝜓) ↔ (¬ 𝜃𝜓))
119, 10bitri 264 . . 3 ((𝜓𝜃) ↔ (¬ 𝜃𝜓))
128, 11sylib 208 . 2 (𝜑 → (¬ 𝜃𝜓))
131, 12mpd 15 1 (𝜑𝜓)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wo 382  w3o 1071
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-or 384  df-3or 1073
This theorem is referenced by:  ivthALT  32632
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