Theorem e222 44633

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

Hypotheses

Ref	Expression
e222.1	⊢ (   𝜑   ,   𝜓   ▶   𝜒   )
e222.2	⊢ (   𝜑   ,   𝜓   ▶   𝜃   )
e222.3	⊢ (   𝜑   ,   𝜓   ▶   𝜏   )
e222.4	⊢ (𝜒 → (𝜃 → (𝜏 → 𝜂)))

Assertion

Ref	Expression
e222	⊢ (   𝜑   ,   𝜓   ▶   𝜂   )

Proof of Theorem e222

Step	Hyp	Ref	Expression
1		e222.3	. . . . . . 7 ⊢ (   𝜑   ,   𝜓   ▶   𝜏   )
2	1	dfvd2i 44582	. . . . . 6 ⊢ (𝜑 → (𝜓 → 𝜏))
3	2	imp 406	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝜏)
4		e222.1	. . . . . . . . 9 ⊢ (   𝜑   ,   𝜓   ▶   𝜒   )
5	4	dfvd2i 44582	. . . . . . . 8 ⊢ (𝜑 → (𝜓 → 𝜒))
6	5	imp 406	. . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → 𝜒)
7		e222.2	. . . . . . . . 9 ⊢ (   𝜑   ,   𝜓   ▶   𝜃   )
8	7	dfvd2i 44582	. . . . . . . 8 ⊢ (𝜑 → (𝜓 → 𝜃))
9	8	imp 406	. . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → 𝜃)
10		e222.4	. . . . . . 7 ⊢ (𝜒 → (𝜃 → (𝜏 → 𝜂)))
11	6, 9, 10	syl2im 40	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → ((𝜑 ∧ 𝜓) → (𝜏 → 𝜂)))
12	11	pm2.43i 52	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → (𝜏 → 𝜂))
13	3, 12	syl5com 31	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → ((𝜑 ∧ 𝜓) → 𝜂))
14	13	pm2.43i 52	. . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝜂)
15	14	ex 412	. 2 ⊢ (𝜑 → (𝜓 → 𝜂))
16	15	dfvd2ir 44583	1 ⊢ (   𝜑   ,   𝜓   ▶   𝜂   )

Syntax hints:   → wi 4   ∧ wa 395  (   wvd2 44574

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 207  df-an 396  df-vd2 44575

This theorem is referenced by:  e220  44634  e202  44636  e022  44638  e002  44640  e020  44642  e200  44644  e221  44646  e212  44648  e122  44650  e112  44651  e121  44653  e211  44654  e22  44668  suctrALT2VD  44832  en3lplem2VD  44840  19.21a3con13vVD  44848  tratrbVD  44857