Theorem ecase2d 1385

Description: Deduction for elimination by cases. (Contributed by NM, 21-Apr-1994.) (Proof shortened by Wolf Lammen, 19-Sep-2024.)

Hypotheses

Ref	Expression
ecase2d.1	⊢ (𝜑 → 𝜓)
ecase2d.2	⊢ (𝜑 → ¬ (𝜓 ∧ 𝜒))
ecase2d.3	⊢ (𝜑 → ¬ (𝜓 ∧ 𝜃))
ecase2d.4	⊢ (𝜑 → (𝜏 ∨ (𝜒 ∨ 𝜃)))

Assertion

Ref	Expression
ecase2d	⊢ (𝜑 → 𝜏)

Proof of Theorem ecase2d

Step	Hyp	Ref	Expression
1		ecase2d.1	. . . 4 ⊢ (𝜑 → 𝜓)
2		ecase2d.2	. . . 4 ⊢ (𝜑 → ¬ (𝜓 ∧ 𝜒))
3	1, 2	mpnanrd 697	. . 3 ⊢ (𝜑 → ¬ 𝜒)
4		ecase2d.3	. . . 4 ⊢ (𝜑 → ¬ (𝜓 ∧ 𝜃))
5	1, 4	mpnanrd 697	. . 3 ⊢ (𝜑 → ¬ 𝜃)
6		ioran 757	. . 3 ⊢ (¬ (𝜒 ∨ 𝜃) ↔ (¬ 𝜒 ∧ ¬ 𝜃))
7	3, 5, 6	sylanbrc 417	. 2 ⊢ (𝜑 → ¬ (𝜒 ∨ 𝜃))
8		ecase2d.4	. 2 ⊢ (𝜑 → (𝜏 ∨ (𝜒 ∨ 𝜃)))
9	7, 8	ecased 1383	1 ⊢ (𝜑 → 𝜏)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ∨ wo 713

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714

This theorem depends on definitions:  df-bi 117

This theorem is referenced by: (None)