Theorem in2an 42117

Description: The virtual deduction introduction rule converting the second conjunct of the second virtual hypothesis into the antecedent of the conclusion. expd 415 is the non-virtual deduction form of in2an 42117. (Contributed by Alan Sare, 30-Jun-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
in2an.1	⊢ (   𝜑   ,   (𝜓 ∧ 𝜒)   ▶   𝜃   )

Assertion

Ref	Expression
in2an	⊢ (   𝜑   ,   𝜓   ▶   (𝜒 → 𝜃)   )

Proof of Theorem in2an

Step	Hyp	Ref	Expression
1		in2an.1	. . . 4 ⊢ (   𝜑   ,   (𝜓 ∧ 𝜒)   ▶   𝜃   )
2	1	dfvd2i 42094	. . 3 ⊢ (𝜑 → ((𝜓 ∧ 𝜒) → 𝜃))
3	2	expd 415	. 2 ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃)))
4	3	dfvd2ir 42095	1 ⊢ (   𝜑   ,   𝜓   ▶   (𝜒 → 𝜃)   )

Syntax hints:   → wi 4   ∧ wa 395  (   wvd2 42086

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 396  df-vd2 42087

This theorem is referenced by:  onfrALTVD  42400