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Theorem ecased 984
Description: Deduction for elimination by cases. (Contributed by NM, 8-Oct-2012.)
Hypotheses
Ref Expression
ecased.1 (𝜑 → (¬ 𝜓𝜃))
ecased.2 (𝜑 → (¬ 𝜒𝜃))
ecased.3 (𝜑 → ((𝜓𝜒) → 𝜃))
Assertion
Ref Expression
ecased (𝜑𝜃)

Proof of Theorem ecased
StepHypRef Expression
1 ecased.1 . 2 (𝜑 → (¬ 𝜓𝜃))
2 ecased.2 . 2 (𝜑 → (¬ 𝜒𝜃))
3 pm3.11 520 . . 3 (¬ (¬ 𝜓 ∨ ¬ 𝜒) → (𝜓𝜒))
4 ecased.3 . . 3 (𝜑 → ((𝜓𝜒) → 𝜃))
53, 4syl5 34 . 2 (𝜑 → (¬ (¬ 𝜓 ∨ ¬ 𝜒) → 𝜃))
61, 2, 5ecase3d 983 1 (𝜑𝜃)
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:  ecase3ad  985  itgsplitioo  23505  rolle  23652  dalaw  34638
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