Theorem ecase2d

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

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

Proof

Step	Hyp	Ref	Expression
1		ecase2d.1	⊢ ( 𝜑 → 𝜓 )
2		ecase2d.2	⊢ ( 𝜑 → ¬ ( 𝜓 ∧ 𝜒 ) )
3		ecase2d.3	⊢ ( 𝜑 → ¬ ( 𝜓 ∧ 𝜃 ) )
4		ecase2d.4	⊢ ( 𝜑 → ( 𝜏 ∨ ( 𝜒 ∨ 𝜃 ) ) )
5	1 2	mpnanrd	⊢ ( 𝜑 → ¬ 𝜒 )
6	1 3	mpnanrd	⊢ ( 𝜑 → ¬ 𝜃 )
7	4	ord	⊢ ( 𝜑 → ( ¬ 𝜏 → ( 𝜒 ∨ 𝜃 ) ) )
8	5 6 7	mtord	⊢ ( 𝜑 → ¬ ¬ 𝜏 )
9	8	notnotrd	⊢ ( 𝜑 → 𝜏 )