Theorem vd12 45044

Description: A virtual deduction with 1 virtual hypothesis virtually inferring a virtual conclusion infers that the same conclusion is virtually inferred by the same virtual hypothesis and an additional hypothesis. (Contributed by Alan Sare, 12-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
vd12.1	⊢ (   𝜑   ▶   𝜓   )

Assertion

Ref	Expression
vd12	⊢ (   𝜑   ,   𝜒   ▶   𝜓   )

Proof of Theorem vd12

Step	Hyp	Ref	Expression
1		vd12.1	. . . 4 ⊢ (   𝜑   ▶   𝜓   )
2	1	in1 45015	. . 3 ⊢ (𝜑 → 𝜓)
3	2	a1d 25	. 2 ⊢ (𝜑 → (𝜒 → 𝜓))
4	3	dfvd2ir 45030	1 ⊢ (   𝜑   ,   𝜒   ▶   𝜓   )

Syntax hints:  (   wvd1 45013  (   wvd2 45021

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 208  df-an 397  df-vd1 45014  df-vd2 45022

This theorem is referenced by:  e221  45093  e212  45095  e122  45097  e112  45098  e121  45100  e211  45101  e120  45107  e12  45167  e21  45173