Theorem vd23 45371

Description: A virtual deduction with 2 virtual hypotheses virtually inferring a virtual conclusion infers that the same conclusion is virtually inferred by the same 2 virtual hypotheses and a third hypothesis. (Contributed by Alan Sare, 12-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
vd23.1	⊢ (   𝜑   ,   𝜓   ▶   𝜒   )

Assertion

Ref	Expression
vd23	⊢ (   𝜑   ,   𝜓   ,   𝜃   ▶   𝜒   )

Proof of Theorem vd23

Step	Hyp	Ref	Expression
1		vd23.1	. . . 4 ⊢ (   𝜑   ,   𝜓   ▶   𝜒   )
2	1	dfvd2i 45354	. . 3 ⊢ (𝜑 → (𝜓 → 𝜒))
3	2	a1dd 51	. 2 ⊢ (𝜑 → (𝜓 → (𝜃 → 𝜒)))
4	3	dfvd3ir 45362	1 ⊢ (   𝜑   ,   𝜓   ,   𝜃   ▶   𝜒   )

This proof depends on syntax axioms:  (   wvd2 45346  (   wvd3 45356

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-3an 1105  df-vd2 45347  df-vd3 45359

This theorem is used by:  e23  45523  e32  45526  e123  45530