Theorem ecase2d

Description: Deduction for elimination by cases. (Contributed by NM, 21-Apr-1994) (Proof shortened by Wolf Lammen, 22-Dec-2012)

	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		idd	⊢ ( 𝜑 → ( 𝜏 → 𝜏 ) )
6	2	pm2.21d	⊢ ( 𝜑 → ( ( 𝜓 ∧ 𝜒 ) → 𝜏 ) )
7	1 6	mpand	⊢ ( 𝜑 → ( 𝜒 → 𝜏 ) )
8	3	pm2.21d	⊢ ( 𝜑 → ( ( 𝜓 ∧ 𝜃 ) → 𝜏 ) )
9	1 8	mpand	⊢ ( 𝜑 → ( 𝜃 → 𝜏 ) )
10	7 9	jaod	⊢ ( 𝜑 → ( ( 𝜒 ∨ 𝜃 ) → 𝜏 ) )
11	5 10 4	mpjaod	⊢ ( 𝜑 → 𝜏 )