Theorem ecase3ad

Description: Deduction for elimination by cases. (Contributed by NM, 24-May-2013) (Proof shortened by Wolf Lammen, 20-Sep-2024)

	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	1	imp	⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝜃 )
5	2	imp	⊢ ( ( 𝜑 ∧ 𝜒 ) → 𝜃 )
6	3	imp	⊢ ( ( 𝜑 ∧ ( ¬ 𝜓 ∧ ¬ 𝜒 ) ) → 𝜃 )
7	4 5 6	pm2.61ddan	⊢ ( 𝜑 → 𝜃 )