Theorem in3an 45348

Description: The virtual deduction introduction rule converting the second conjunct of the third virtual hypothesis into the antecedent of the conclusion. exp4a 436 is the non-virtual deduction form of in3an 45348. (Contributed by Alan Sare, 25-Jun-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
in3an.1	⊢ (   𝜑   ,   𝜓   ,   (𝜒 ∧ 𝜃)   ▶   𝜏   )

Assertion

Ref	Expression
in3an	⊢ (   𝜑   ,   𝜓   ,   𝜒   ▶   (𝜃 → 𝜏)   )

Proof of Theorem in3an

Step	Hyp	Ref	Expression
1		in3an.1	. . . 4 ⊢ (   𝜑   ,   𝜓   ,   (𝜒 ∧ 𝜃)   ▶   𝜏   )
2	1	dfvd3i 45329	. . 3 ⊢ (𝜑 → (𝜓 → ((𝜒 ∧ 𝜃) → 𝜏)))
3	2	exp4a 436	. 2 ⊢ (𝜑 → (𝜓 → (𝜒 → (𝜃 → 𝜏))))
4	3	dfvd3ir 45330	1 ⊢ (   𝜑   ,   𝜓   ,   𝜒   ▶   (𝜃 → 𝜏)   )

This proof depends on syntax axioms:   → wi 4   ∧ wa 400  (   wvd3 45324

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-an 401  df-3an 1105  df-vd3 45327

This theorem is used by:  onfrALTlem2VD  45625