Theorem e13 45516

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 45370	. 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 45502	1 ⊢ (   𝜑   ,   𝜒   ,   𝜃   ▶   𝜂   )

This proof depends on syntax axioms:   → wi 4  (   wvd1 45338  (   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-vd1 45339  df-vd3 45359

This theorem is used by:  e13an  45517  en3lplem2VD  45612  rspsbc2VD  45623  ssralv2VD  45634  imbi12VD  45641  imbi13VD  45642  truniALTVD  45646