Theorem e222 39178

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 39118	. . . . . 6 ⊢ (𝜑 → (𝜓 → 𝜏))
3	2	imp 444	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝜏)
4		e222.1	. . . . . . . . 9 ⊢ (   𝜑   ,   𝜓   ▶   𝜒   )
5	4	dfvd2i 39118	. . . . . . . 8 ⊢ (𝜑 → (𝜓 → 𝜒))
6	5	imp 444	. . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → 𝜒)
7		e222.2	. . . . . . . . 9 ⊢ (   𝜑   ,   𝜓   ▶   𝜃   )
8	7	dfvd2i 39118	. . . . . . . 8 ⊢ (𝜑 → (𝜓 → 𝜃))
9	8	imp 444	. . . . . . 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 449	. 2 ⊢ (𝜑 → (𝜓 → 𝜂))
16	15	dfvd2ir 39119	1 ⊢ (   𝜑   ,   𝜓   ▶   𝜂   )

Syntax hints:   → wi 4   ∧ wa 383  (   wvd2 39110

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 197  df-an 385  df-vd2 39111

This theorem is referenced by:  e220  39179  e202  39181  e022  39183  e002  39185  e020  39187  e200  39189  e221  39191  e212  39193  e122  39195  e112  39196  e121  39198  e211  39199  e22  39213  suctrALT2VD  39385  en3lplem2VD  39393  19.21a3con13vVD  39401  tratrbVD  39411