Theorem e222 41342

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 41291	. . . . . 6 ⊢ (𝜑 → (𝜓 → 𝜏))
3	2	imp 410	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝜏)
4		e222.1	. . . . . . . . 9 ⊢ (   𝜑   ,   𝜓   ▶   𝜒   )
5	4	dfvd2i 41291	. . . . . . . 8 ⊢ (𝜑 → (𝜓 → 𝜒))
6	5	imp 410	. . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → 𝜒)
7		e222.2	. . . . . . . . 9 ⊢ (   𝜑   ,   𝜓   ▶   𝜃   )
8	7	dfvd2i 41291	. . . . . . . 8 ⊢ (𝜑 → (𝜓 → 𝜃))
9	8	imp 410	. . . . . . 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 416	. 2 ⊢ (𝜑 → (𝜓 → 𝜂))
16	15	dfvd2ir 41292	1 ⊢ (   𝜑   ,   𝜓   ▶   𝜂   )

Syntax hints:   → wi 4   ∧ wa 399  (   wvd2 41283

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 210  df-an 400  df-vd2 41284

This theorem is referenced by:  e220  41343  e202  41345  e022  41347  e002  41349  e020  41351  e200  41353  e221  41355  e212  41357  e122  41359  e112  41360  e121  41362  e211  41363  e22  41377  suctrALT2VD  41542  en3lplem2VD  41550  19.21a3con13vVD  41558  tratrbVD  41567