Theorem in3an 42231

Description: The virtual deduction introduction rule converting the second conjunct of the third virtual hypothesis into the antecedent of the conclusion. exp4a 432 is the non-virtual deduction form of in3an 42231. (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 42212	. . 3 ⊢ (𝜑 → (𝜓 → ((𝜒 ∧ 𝜃) → 𝜏)))
3	2	exp4a 432	. 2 ⊢ (𝜑 → (𝜓 → (𝜒 → (𝜃 → 𝜏))))
4	3	dfvd3ir 42213	1 ⊢ (   𝜑   ,   𝜓   ,   𝜒   ▶   (𝜃 → 𝜏)   )

Syntax hints:   → wi 4   ∧ wa 396  (   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:  onfrALTlem2VD  42509