Theorem e13 42257

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

Hypotheses

Ref	Expression
e13.1	⊢ (   𝜑   ▶   𝜓   )
e13.2	⊢ (   𝜑   ,   𝜒   ,   𝜃   ▶   𝜏   )
e13.3	⊢ (𝜓 → (𝜏 → 𝜂))

Assertion

Ref	Expression
e13	⊢ (   𝜑   ,   𝜒   ,   𝜃   ▶   𝜂   )

Proof of Theorem e13

Step	Hyp	Ref	Expression
1		e13.1	. . 3 ⊢ (   𝜑   ▶   𝜓   )
2	1	vd13 42110	. 2 ⊢ (   𝜑   ,   𝜒   ,   𝜃   ▶   𝜓   )
3		e13.2	. 2 ⊢ (   𝜑   ,   𝜒   ,   𝜃   ▶   𝜏   )
4		e13.3	. 2 ⊢ (𝜓 → (𝜏 → 𝜂))
5	2, 3, 4	e33 42243	1 ⊢ (   𝜑   ,   𝜒   ,   𝜃   ▶   𝜂   )

Syntax hints:   → wi 4  (   wvd1 42078  (   wvd3 42096

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 396  df-3an 1087  df-vd1 42079  df-vd3 42099

This theorem is referenced by:  e13an  42258  en3lplem2VD  42353  rspsbc2VD  42364  ssralv2VD  42375  imbi12VD  42382  imbi13VD  42383  truniALTVD  42387