Theorem dfvd3 42211

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

Assertion

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

Proof of Theorem dfvd3

Step	Hyp	Ref	Expression
1		df-vd3 42210	. 2 ⊢ ((   𝜑   ,   𝜓   ,   𝜒   ▶   𝜃   ) ↔ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃))
2		df-3an 1088	. . . . 5 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ((𝜑 ∧ 𝜓) ∧ 𝜒))
3	2	imbi1i 350	. . . 4 ⊢ (((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) ↔ (((𝜑 ∧ 𝜓) ∧ 𝜒) → 𝜃))
4		impexp 451	. . . 4 ⊢ ((((𝜑 ∧ 𝜓) ∧ 𝜒) → 𝜃) ↔ ((𝜑 ∧ 𝜓) → (𝜒 → 𝜃)))
5	3, 4	bitri 274	. . 3 ⊢ (((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) ↔ ((𝜑 ∧ 𝜓) → (𝜒 → 𝜃)))
6		impexp 451	. . 3 ⊢ (((𝜑 ∧ 𝜓) → (𝜒 → 𝜃)) ↔ (𝜑 → (𝜓 → (𝜒 → 𝜃))))
7	5, 6	bitri 274	. 2 ⊢ (((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) ↔ (𝜑 → (𝜓 → (𝜒 → 𝜃))))
8	1, 7	bitri 274	1 ⊢ ((   𝜑   ,   𝜓   ,   𝜒   ▶   𝜃   ) ↔ (𝜑 → (𝜓 → (𝜒 → 𝜃))))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1086  (   wvd3 42207

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 206  df-an 397  df-3an 1088  df-vd3 42210

This theorem is referenced by:  dfvd3i  42212  dfvd3ir  42213