Theorem e223 42255

Description: A virtual deduction elimination rule. (Contributed by Alan Sare, 12-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
e223.1	⊢ (   𝜑   ,   𝜓   ▶   𝜒   )
e223.2	⊢ (   𝜑   ,   𝜓   ▶   𝜃   )
e223.3	⊢ (   𝜑   ,   𝜓   ,   𝜏   ▶   𝜂   )
e223.4	⊢ (𝜒 → (𝜃 → (𝜂 → 𝜁)))

Assertion

Ref	Expression
e223	⊢ (   𝜑   ,   𝜓   ,   𝜏   ▶   𝜁   )

Proof of Theorem e223

Step	Hyp	Ref	Expression
1		e223.1	. . . . 5 ⊢ (   𝜑   ,   𝜓   ▶   𝜒   )
2	1	in2 42225	. . . 4 ⊢ (   𝜑   ▶   (𝜓 → 𝜒)   )
3	2	in1 42191	. . 3 ⊢ (𝜑 → (𝜓 → 𝜒))
4		e223.2	. . . . 5 ⊢ (   𝜑   ,   𝜓   ▶   𝜃   )
5	4	in2 42225	. . . 4 ⊢ (   𝜑   ▶   (𝜓 → 𝜃)   )
6	5	in1 42191	. . 3 ⊢ (𝜑 → (𝜓 → 𝜃))
7		e223.3	. . . . . 6 ⊢ (   𝜑   ,   𝜓   ,   𝜏   ▶   𝜂   )
8	7	in3 42229	. . . . 5 ⊢ (   𝜑   ,   𝜓   ▶   (𝜏 → 𝜂)   )
9	8	in2 42225	. . . 4 ⊢ (   𝜑   ▶   (𝜓 → (𝜏 → 𝜂))   )
10	9	in1 42191	. . 3 ⊢ (𝜑 → (𝜓 → (𝜏 → 𝜂)))
11		e223.4	. . 3 ⊢ (𝜒 → (𝜃 → (𝜂 → 𝜁)))
12	3, 6, 10, 11	ee223 42254	. 2 ⊢ (𝜑 → (𝜓 → (𝜏 → 𝜁)))
13	12	dfvd3ir 42213	1 ⊢ (   𝜑   ,   𝜓   ,   𝜏   ▶   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-vd1 42190  df-vd2 42198  df-vd3 42210

This theorem is referenced by:  tratrbVD  42481