Theorem e222 45373

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 45322	. . . . . 6 ⊢ (𝜑 → (𝜓 → 𝜏))
3	2	imp 411	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝜏)
4		e222.1	. . . . . . . . 9 ⊢ (   𝜑   ,   𝜓   ▶   𝜒   )
5	4	dfvd2i 45322	. . . . . . . 8 ⊢ (𝜑 → (𝜓 → 𝜒))
6	5	imp 411	. . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → 𝜒)
7		e222.2	. . . . . . . . 9 ⊢ (   𝜑   ,   𝜓   ▶   𝜃   )
8	7	dfvd2i 45322	. . . . . . . 8 ⊢ (𝜑 → (𝜓 → 𝜃))
9	8	imp 411	. . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → 𝜃)
10		e222.4	. . . . . . 7 ⊢ (𝜒 → (𝜃 → (𝜏 → 𝜂)))
11	6, 9, 10	syl2im 41	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → ((𝜑 ∧ 𝜓) → (𝜏 → 𝜂)))
12	11	pm2.43i 53	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → (𝜏 → 𝜂))
13	3, 12	syl5com 32	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → ((𝜑 ∧ 𝜓) → 𝜂))
14	13	pm2.43i 53	. . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝜂)
15	14	ex 417	. 2 ⊢ (𝜑 → (𝜓 → 𝜂))
16	15	dfvd2ir 45323	1 ⊢ (   𝜑   ,   𝜓   ▶   𝜂   )

This proof depends on syntax axioms:   → wi 4   ∧ wa 400  (   wvd2 45314

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-an 401  df-vd2 45315

This theorem is used by:  e220  45374  e202  45376  e022  45378  e002  45380  e020  45382  e200  45384  e221  45386  e212  45388  e122  45390  e112  45391  e121  45393  e211  45394  e22  45408  suctrALT2VD  45572  en3lplem2VD  45580  19.21a3con13vVD  45588  tratrbVD  45597