Theorem ecase13d

Description: Deduction for elimination by cases. (Contributed by Jeff Hankins, 18-Aug-2009)

	Ref	Expression
Hypotheses	ecase13d.1	⊢ ( 𝜑 → ¬ 𝜒 )
	ecase13d.2	⊢ ( 𝜑 → ¬ 𝜃 )
	ecase13d.3	⊢ ( 𝜑 → ( 𝜒 ∨ 𝜓 ∨ 𝜃 ) )
Assertion	ecase13d	⊢ ( 𝜑 → 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		ecase13d.1	⊢ ( 𝜑 → ¬ 𝜒 )
2		ecase13d.2	⊢ ( 𝜑 → ¬ 𝜃 )
3		ecase13d.3	⊢ ( 𝜑 → ( 𝜒 ∨ 𝜓 ∨ 𝜃 ) )
4		3orass	⊢ ( ( 𝜒 ∨ 𝜓 ∨ 𝜃 ) ↔ ( 𝜒 ∨ ( 𝜓 ∨ 𝜃 ) ) )
5		df-or	⊢ ( ( 𝜒 ∨ ( 𝜓 ∨ 𝜃 ) ) ↔ ( ¬ 𝜒 → ( 𝜓 ∨ 𝜃 ) ) )
6	4 5	bitri	⊢ ( ( 𝜒 ∨ 𝜓 ∨ 𝜃 ) ↔ ( ¬ 𝜒 → ( 𝜓 ∨ 𝜃 ) ) )
7	3 6	sylib	⊢ ( 𝜑 → ( ¬ 𝜒 → ( 𝜓 ∨ 𝜃 ) ) )
8	1 7	mpd	⊢ ( 𝜑 → ( 𝜓 ∨ 𝜃 ) )
9		orcom	⊢ ( ( 𝜓 ∨ 𝜃 ) ↔ ( 𝜃 ∨ 𝜓 ) )
10		df-or	⊢ ( ( 𝜃 ∨ 𝜓 ) ↔ ( ¬ 𝜃 → 𝜓 ) )
11	9 10	bitri	⊢ ( ( 𝜓 ∨ 𝜃 ) ↔ ( ¬ 𝜃 → 𝜓 ) )
12	8 11	sylib	⊢ ( 𝜑 → ( ¬ 𝜃 → 𝜓 ) )
13	2 12	mpd	⊢ ( 𝜑 → 𝜓 )