Theorem vd13 42221

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 a two additional hypotheses. (Contributed by Alan Sare, 12-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
vd13.1	⊢ (   𝜑   ▶   𝜓   )

Assertion

Ref	Expression
vd13	⊢ (   𝜑   ,   𝜒   ,   𝜃   ▶   𝜓   )

Proof of Theorem vd13

Step	Hyp	Ref	Expression
1		vd13.1	. . . . 5 ⊢ (   𝜑   ▶   𝜓   )
2	1	in1 42191	. . . 4 ⊢ (𝜑 → 𝜓)
3	2	a1d 25	. . 3 ⊢ (𝜑 → (𝜒 → 𝜓))
4	3	a1dd 50	. 2 ⊢ (𝜑 → (𝜒 → (𝜃 → 𝜓)))
5	4	dfvd3ir 42213	1 ⊢ (   𝜑   ,   𝜒   ,   𝜃   ▶   𝜓   )

Syntax hints:  (   wvd1 42189  (   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-vd1 42190  df-vd3 42210

This theorem is referenced by:  e13  42368  e31  42371  e123  42382