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Theorem e223 37881
 Description: A virtual deduction elimination rule. (Contributed by Alan Sare, 12-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
e223.1 (   𝜑   ,   𝜓   ▶   𝜒   )
e223.2 (   𝜑   ,   𝜓   ▶   𝜃   )
e223.3 (   𝜑   ,   𝜓   ,   𝜏   ▶   𝜂   )
e223.4 (𝜒 → (𝜃 → (𝜂𝜁)))
Assertion
Ref Expression
e223 (   𝜑   ,   𝜓   ,   𝜏   ▶   𝜁   )

Proof of Theorem e223
StepHypRef Expression
1 e223.1 . . . . 5 (   𝜑   ,   𝜓   ▶   𝜒   )
21in2 37851 . . . 4 (   𝜑   ▶   (𝜓𝜒)   )
32in1 37808 . . 3 (𝜑 → (𝜓𝜒))
4 e223.2 . . . . 5 (   𝜑   ,   𝜓   ▶   𝜃   )
54in2 37851 . . . 4 (   𝜑   ▶   (𝜓𝜃)   )
65in1 37808 . . 3 (𝜑 → (𝜓𝜃))
7 e223.3 . . . . . 6 (   𝜑   ,   𝜓   ,   𝜏   ▶   𝜂   )
87in3 37855 . . . . 5 (   𝜑   ,   𝜓   ▶   (𝜏𝜂)   )
98in2 37851 . . . 4 (   𝜑   ▶   (𝜓 → (𝜏𝜂))   )
109in1 37808 . . 3 (𝜑 → (𝜓 → (𝜏𝜂)))
11 e223.4 . . 3 (𝜒 → (𝜃 → (𝜂𝜁)))
123, 6, 10, 11ee223 37880 . 2 (𝜑 → (𝜓 → (𝜏𝜁)))
1312dfvd3ir 37830 1 (   𝜑   ,   𝜓   ,   𝜏   ▶   𝜁   )
 Colors of variables: wff setvar class Syntax hints:   → wi 4  (   wvd2 37814  (   wvd3 37824 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 196  df-an 385  df-3an 1033  df-vd1 37807  df-vd2 37815  df-vd3 37827 This theorem is referenced by:  tratrbVD  38119
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