Theorem vd12 45582

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 45553	. . 3 ⊢ (𝜑 → 𝜓)
3	2	a1d 26	. 2 ⊢ (𝜑 → (𝜒 → 𝜓))
4	3	dfvd2ir 45568	1 ⊢ (   𝜑   ,   𝜒   ▶   𝜓   )

This proof depends on syntax axioms:  (   wvd1 45551  (   wvd2 45559

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 402  df-vd1 45552  df-vd2 45560

This theorem is used by:  e221  45631  e212  45633  e122  45635  e112  45636  e121  45638  e211  45639  e120  45645  e12  45705  e21  45711