Theorem e31 42371

Description: A virtual deduction elimination rule. (Contributed by Alan Sare, 13-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
e31.1	⊢ (   𝜑   ,   𝜓   ,   𝜒   ▶   𝜃   )
e31.2	⊢ (   𝜑   ▶   𝜏   )
e31.3	⊢ (𝜃 → (𝜏 → 𝜂))

Assertion

Ref	Expression
e31	⊢ (   𝜑   ,   𝜓   ,   𝜒   ▶   𝜂   )

Proof of Theorem e31

Step	Hyp	Ref	Expression
1		e31.1	. 2 ⊢ (   𝜑   ,   𝜓   ,   𝜒   ▶   𝜃   )
2		e31.2	. . 3 ⊢ (   𝜑   ▶   𝜏   )
3	2	vd13 42221	. 2 ⊢ (   𝜑   ,   𝜓   ,   𝜒   ▶   𝜏   )
4		e31.3	. 2 ⊢ (𝜃 → (𝜏 → 𝜂))
5	1, 3, 4	e33 42354	1 ⊢ (   𝜑   ,   𝜓   ,   𝜒   ▶   𝜂   )

Syntax hints:   → wi 4  (   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:  e31an  42373  trintALTVD  42500