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Theorem ecase2d 980
Description: Deduction for elimination by cases. (Contributed by NM, 21-Apr-1994.) (Proof shortened by Wolf Lammen, 22-Dec-2012.)
Hypotheses
Ref Expression
ecase2d.1 (𝜑𝜓)
ecase2d.2 (𝜑 → ¬ (𝜓𝜒))
ecase2d.3 (𝜑 → ¬ (𝜓𝜃))
ecase2d.4 (𝜑 → (𝜏 ∨ (𝜒𝜃)))
Assertion
Ref Expression
ecase2d (𝜑𝜏)

Proof of Theorem ecase2d
StepHypRef Expression
1 idd 24 . 2 (𝜑 → (𝜏𝜏))
2 ecase2d.1 . . . 4 (𝜑𝜓)
3 ecase2d.2 . . . . 5 (𝜑 → ¬ (𝜓𝜒))
43pm2.21d 118 . . . 4 (𝜑 → ((𝜓𝜒) → 𝜏))
52, 4mpand 710 . . 3 (𝜑 → (𝜒𝜏))
6 ecase2d.3 . . . . 5 (𝜑 → ¬ (𝜓𝜃))
76pm2.21d 118 . . . 4 (𝜑 → ((𝜓𝜃) → 𝜏))
82, 7mpand 710 . . 3 (𝜑 → (𝜃𝜏))
95, 8jaod 395 . 2 (𝜑 → ((𝜒𝜃) → 𝜏))
10 ecase2d.4 . 2 (𝜑 → (𝜏 ∨ (𝜒𝜃)))
111, 9, 10mpjaod 396 1 (𝜑𝜏)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wo 383  wa 384
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 385  df-an 386
This theorem is referenced by: (None)
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