Theorem e1111 42295

Description: A virtual deduction elimination rule. (Contributed by Alan Sare, 6-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
e1111.1	⊢ (   𝜑   ▶   𝜓   )
e1111.2	⊢ (   𝜑   ▶   𝜒   )
e1111.3	⊢ (   𝜑   ▶   𝜃   )
e1111.4	⊢ (   𝜑   ▶   𝜏   )
e1111.5	⊢ (𝜓 → (𝜒 → (𝜃 → (𝜏 → 𝜂))))

Assertion

Ref	Expression
e1111	⊢ (   𝜑   ▶   𝜂   )

Proof of Theorem e1111

Step	Hyp	Ref	Expression
1		e1111.1	. . . 4 ⊢ (   𝜑   ▶   𝜓   )
2	1	in1 42191	. . 3 ⊢ (𝜑 → 𝜓)
3		e1111.2	. . . 4 ⊢ (   𝜑   ▶   𝜒   )
4	3	in1 42191	. . 3 ⊢ (𝜑 → 𝜒)
5		e1111.3	. . . 4 ⊢ (   𝜑   ▶   𝜃   )
6	5	in1 42191	. . 3 ⊢ (𝜑 → 𝜃)
7		e1111.4	. . . 4 ⊢ (   𝜑   ▶   𝜏   )
8	7	in1 42191	. . 3 ⊢ (𝜑 → 𝜏)
9		e1111.5	. . 3 ⊢ (𝜓 → (𝜒 → (𝜃 → (𝜏 → 𝜂))))
10	2, 4, 6, 8, 9	ee1111 42136	. 2 ⊢ (𝜑 → 𝜂)
11	10	dfvd1ir 42193	1 ⊢ (   𝜑   ▶   𝜂   )

Syntax hints:   → wi 4  (   wvd1 42189

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-vd1 42190

This theorem is referenced by:  trsbcVD  42497