Theorem e323 42386

Description: A virtual deduction elimination rule. (Contributed by Alan Sare, 17-Apr-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
e323.1	⊢ (   𝜑   ,   𝜓   ,   𝜒   ▶   𝜃   )
e323.2	⊢ (   𝜑   ,   𝜓   ▶   𝜏   )
e323.3	⊢ (   𝜑   ,   𝜓   ,   𝜒   ▶   𝜂   )
e323.4	⊢ (𝜃 → (𝜏 → (𝜂 → 𝜁)))

Assertion

Ref	Expression
e323	⊢ (   𝜑   ,   𝜓   ,   𝜒   ▶   𝜁   )

Proof of Theorem e323

Step	Hyp	Ref	Expression
1		e323.1	. . . 4 ⊢ (   𝜑   ,   𝜓   ,   𝜒   ▶   𝜃   )
2	1	dfvd3i 42212	. . 3 ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃)))
3		e323.2	. . . 4 ⊢ (   𝜑   ,   𝜓   ▶   𝜏   )
4	3	dfvd2i 42205	. . 3 ⊢ (𝜑 → (𝜓 → 𝜏))
5		e323.3	. . . 4 ⊢ (   𝜑   ,   𝜓   ,   𝜒   ▶   𝜂   )
6	5	dfvd3i 42212	. . 3 ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜂)))
7		e323.4	. . 3 ⊢ (𝜃 → (𝜏 → (𝜂 → 𝜁)))
8	2, 4, 6, 7	ee323 42128	. 2 ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜁)))
9	8	dfvd3ir 42213	1 ⊢ (   𝜑   ,   𝜓   ,   𝜒   ▶   𝜁   )

Syntax hints:   → wi 4  (   wvd2 42197  (   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-vd2 42198  df-vd3 42210

This theorem is referenced by:  trintALTVD  42500