Theorem ecase3ad

Description: Deduction for elimination by cases. (Contributed by NM, 24-May-2013)

	Ref	Expression
Hypotheses	ecase3ad.1	⊢ ( 𝜑 → ( 𝜓 → 𝜃 ) )
	ecase3ad.2	⊢ ( 𝜑 → ( 𝜒 → 𝜃 ) )
	ecase3ad.3	⊢ ( 𝜑 → ( ( ¬ 𝜓 ∧ ¬ 𝜒 ) → 𝜃 ) )
Assertion	ecase3ad	⊢ ( 𝜑 → 𝜃 )

Proof

Step	Hyp	Ref	Expression
1		ecase3ad.1	⊢ ( 𝜑 → ( 𝜓 → 𝜃 ) )
2		ecase3ad.2	⊢ ( 𝜑 → ( 𝜒 → 𝜃 ) )
3		ecase3ad.3	⊢ ( 𝜑 → ( ( ¬ 𝜓 ∧ ¬ 𝜒 ) → 𝜃 ) )
4		notnotr	⊢ ( ¬ ¬ 𝜓 → 𝜓 )
5	4 1	syl5	⊢ ( 𝜑 → ( ¬ ¬ 𝜓 → 𝜃 ) )
6		notnotr	⊢ ( ¬ ¬ 𝜒 → 𝜒 )
7	6 2	syl5	⊢ ( 𝜑 → ( ¬ ¬ 𝜒 → 𝜃 ) )
8	5 7 3	ecased	⊢ ( 𝜑 → 𝜃 )