Theorem e222 44599

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 44548	. . . . . 6 ⊢ (𝜑 → (𝜓 → 𝜏))
3	2	imp 406	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝜏)
4		e222.1	. . . . . . . . 9 ⊢ (   𝜑   ,   𝜓   ▶   𝜒   )
5	4	dfvd2i 44548	. . . . . . . 8 ⊢ (𝜑 → (𝜓 → 𝜒))
6	5	imp 406	. . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → 𝜒)
7		e222.2	. . . . . . . . 9 ⊢ (   𝜑   ,   𝜓   ▶   𝜃   )
8	7	dfvd2i 44548	. . . . . . . 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 44549	1 ⊢ (   𝜑   ,   𝜓   ▶   𝜂   )

Syntax hints:   → wi 4   ∧ wa 395  (   wvd2 44540

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 44541

This theorem is referenced by:  e220  44600  e202  44602  e022  44604  e002  44606  e020  44608  e200  44610  e221  44612  e212  44614  e122  44616  e112  44617  e121  44619  e211  44620  e22  44634  suctrALT2VD  44798  en3lplem2VD  44806  19.21a3con13vVD  44814  tratrbVD  44823