Theorem ecase23d

Description: Deduction for elimination by cases. (Contributed by NM, 22-Apr-1994)

	Ref	Expression
Hypotheses	ecase23d.1	⊢ ( 𝜑 → ¬ 𝜒 )
	ecase23d.2	⊢ ( 𝜑 → ¬ 𝜃 )
	ecase23d.3	⊢ ( 𝜑 → ( 𝜓 ∨ 𝜒 ∨ 𝜃 ) )
Assertion	ecase23d	⊢ ( 𝜑 → 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		ecase23d.1	⊢ ( 𝜑 → ¬ 𝜒 )
2		ecase23d.2	⊢ ( 𝜑 → ¬ 𝜃 )
3		ecase23d.3	⊢ ( 𝜑 → ( 𝜓 ∨ 𝜒 ∨ 𝜃 ) )
4		3orass	⊢ ( ( 𝜓 ∨ 𝜒 ∨ 𝜃 ) ↔ ( 𝜓 ∨ ( 𝜒 ∨ 𝜃 ) ) )
5	3 4	sylib	⊢ ( 𝜑 → ( 𝜓 ∨ ( 𝜒 ∨ 𝜃 ) ) )
6		ioran	⊢ ( ¬ ( 𝜒 ∨ 𝜃 ) ↔ ( ¬ 𝜒 ∧ ¬ 𝜃 ) )
7	1 2 6	sylanbrc	⊢ ( 𝜑 → ¬ ( 𝜒 ∨ 𝜃 ) )
8	5 7	olcnd	⊢ ( 𝜑 → 𝜓 )