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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 → (𝜏𝜂)))
Assertion
Ref Expression
e222 (   𝜑   ,   𝜓   ▶   𝜂   )

Proof of Theorem e222
StepHypRef Expression
1 e222.3 . . . . . . 7 (   𝜑   ,   𝜓   ▶   𝜏   )
21dfvd2i 39118 . . . . . 6 (𝜑 → (𝜓𝜏))
32imp 444 . . . . 5 ((𝜑𝜓) → 𝜏)
4 e222.1 . . . . . . . . 9 (   𝜑   ,   𝜓   ▶   𝜒   )
54dfvd2i 39118 . . . . . . . 8 (𝜑 → (𝜓𝜒))
65imp 444 . . . . . . 7 ((𝜑𝜓) → 𝜒)
7 e222.2 . . . . . . . . 9 (   𝜑   ,   𝜓   ▶   𝜃   )
87dfvd2i 39118 . . . . . . . 8 (𝜑 → (𝜓𝜃))
98imp 444 . . . . . . 7 ((𝜑𝜓) → 𝜃)
10 e222.4 . . . . . . 7 (𝜒 → (𝜃 → (𝜏𝜂)))
116, 9, 10syl2im 40 . . . . . 6 ((𝜑𝜓) → ((𝜑𝜓) → (𝜏𝜂)))
1211pm2.43i 52 . . . . 5 ((𝜑𝜓) → (𝜏𝜂))
133, 12syl5com 31 . . . 4 ((𝜑𝜓) → ((𝜑𝜓) → 𝜂))
1413pm2.43i 52 . . 3 ((𝜑𝜓) → 𝜂)
1514ex 449 . 2 (𝜑 → (𝜓𝜂))
1615dfvd2ir 39119 1 (   𝜑   ,   𝜓   ▶   𝜂   )
Colors of variables: wff setvar class
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
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