Theorem ecased 1031

Description: Deduction for elimination by cases. (Contributed by NM, 8-Oct-2012.)

Hypotheses

Ref	Expression
ecased.1	⊢ (𝜑 → (¬ 𝜓 → 𝜃))
ecased.2	⊢ (𝜑 → (¬ 𝜒 → 𝜃))
ecased.3	⊢ (𝜑 → ((𝜓 ∧ 𝜒) → 𝜃))

Assertion

Ref	Expression
ecased	⊢ (𝜑 → 𝜃)

Proof of Theorem ecased

Step	Hyp	Ref	Expression
1		ecased.1	. 2 ⊢ (𝜑 → (¬ 𝜓 → 𝜃))
2		ecased.2	. 2 ⊢ (𝜑 → (¬ 𝜒 → 𝜃))
3		pm3.11 989	. . 3 ⊢ (¬ (¬ 𝜓 ∨ ¬ 𝜒) → (𝜓 ∧ 𝜒))
4		ecased.3	. . 3 ⊢ (𝜑 → ((𝜓 ∧ 𝜒) → 𝜃))
5	3, 4	syl5 34	. 2 ⊢ (𝜑 → (¬ (¬ 𝜓 ∨ ¬ 𝜒) → 𝜃))
6	1, 2, 5	ecase3d 1030	1 ⊢ (𝜑 → 𝜃)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∨ wo 843

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 206  df-an 396  df-or 844

This theorem is referenced by:  ecase3adOLD  1033  itgsplitioo  24983  rolle  25135  dalaw  37879