Theorem dfvd3

Description: Definition of a 3-hypothesis virtual deduction. (Contributed by Alan Sare, 14-Nov-2011) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	dfvd3	⊢ ( (    𝜑    ,    𝜓    ,    𝜒    ▶    𝜃    ) ↔ ( 𝜑 → ( 𝜓 → ( 𝜒 → 𝜃 ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		df-vd3	⊢ ( (    𝜑    ,    𝜓    ,    𝜒    ▶    𝜃    ) ↔ ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) → 𝜃 ) )
2		df-3an	⊢ ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) ↔ ( ( 𝜑 ∧ 𝜓 ) ∧ 𝜒 ) )
3	2	imbi1i	⊢ ( ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) → 𝜃 ) ↔ ( ( ( 𝜑 ∧ 𝜓 ) ∧ 𝜒 ) → 𝜃 ) )
4		impexp	⊢ ( ( ( ( 𝜑 ∧ 𝜓 ) ∧ 𝜒 ) → 𝜃 ) ↔ ( ( 𝜑 ∧ 𝜓 ) → ( 𝜒 → 𝜃 ) ) )
5	3 4	bitri	⊢ ( ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) → 𝜃 ) ↔ ( ( 𝜑 ∧ 𝜓 ) → ( 𝜒 → 𝜃 ) ) )
6		impexp	⊢ ( ( ( 𝜑 ∧ 𝜓 ) → ( 𝜒 → 𝜃 ) ) ↔ ( 𝜑 → ( 𝜓 → ( 𝜒 → 𝜃 ) ) ) )
7	5 6	bitri	⊢ ( ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) → 𝜃 ) ↔ ( 𝜑 → ( 𝜓 → ( 𝜒 → 𝜃 ) ) ) )
8	1 7	bitri	⊢ ( (    𝜑    ,    𝜓    ,    𝜒    ▶    𝜃    ) ↔ ( 𝜑 → ( 𝜓 → ( 𝜒 → 𝜃 ) ) ) )