Theorem dfvd3an 45576

Description: Definition of a 3-hypothesis virtual deduction in vd conjunction form. (Contributed by Alan Sare, 13-Jun-2015.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
dfvd3an	⊢ ((   (   𝜑   ,   𝜓   ,   𝜒   )   ▶   𝜃   ) ↔ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃))

Proof of Theorem dfvd3an

Step	Hyp	Ref	Expression
1		df-vd1 45552	. 2 ⊢ ((   (   𝜑   ,   𝜓   ,   𝜒   )   ▶   𝜃   ) ↔ ((   𝜑   ,   𝜓   ,   𝜒   ) → 𝜃))
2		df-vhc3 45571	. . 3 ⊢ ((   𝜑   ,   𝜓   ,   𝜒   ) ↔ (𝜑 ∧ 𝜓 ∧ 𝜒))
3	2	imbi1i 352	. 2 ⊢ (((   𝜑   ,   𝜓   ,   𝜒   ) → 𝜃) ↔ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃))
4	1, 3	bitri 278	1 ⊢ ((   (   𝜑   ,   𝜓   ,   𝜒   )   ▶   𝜃   ) ↔ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ w3a 1103  (   wvd1 45551  (   wvhc3 45570

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-vd1 45552  df-vhc3 45571

This theorem is used by:  dfvd3ani  45577  dfvd3anir  45578