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Theorem e333 37981
 Description: A virtual deduction elimination rule. (Contributed by Alan Sare, 12-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
e333.1 (   𝜑   ,   𝜓   ,   𝜒   ▶   𝜃   )
e333.2 (   𝜑   ,   𝜓   ,   𝜒   ▶   𝜏   )
e333.3 (   𝜑   ,   𝜓   ,   𝜒   ▶   𝜂   )
e333.4 (𝜃 → (𝜏 → (𝜂𝜁)))
Assertion
Ref Expression
e333 (   𝜑   ,   𝜓   ,   𝜒   ▶   𝜁   )

Proof of Theorem e333
StepHypRef Expression
1 e333.3 . . . . . . 7 (   𝜑   ,   𝜓   ,   𝜒   ▶   𝜂   )
21dfvd3i 37829 . . . . . 6 (𝜑 → (𝜓 → (𝜒𝜂)))
323imp 1249 . . . . 5 ((𝜑𝜓𝜒) → 𝜂)
4 e333.1 . . . . . . . . 9 (   𝜑   ,   𝜓   ,   𝜒   ▶   𝜃   )
54dfvd3i 37829 . . . . . . . 8 (𝜑 → (𝜓 → (𝜒𝜃)))
653imp 1249 . . . . . . 7 ((𝜑𝜓𝜒) → 𝜃)
7 e333.2 . . . . . . . . 9 (   𝜑   ,   𝜓   ,   𝜒   ▶   𝜏   )
87dfvd3i 37829 . . . . . . . 8 (𝜑 → (𝜓 → (𝜒𝜏)))
983imp 1249 . . . . . . 7 ((𝜑𝜓𝜒) → 𝜏)
10 e333.4 . . . . . . 7 (𝜃 → (𝜏 → (𝜂𝜁)))
116, 9, 10syl2im 39 . . . . . 6 ((𝜑𝜓𝜒) → ((𝜑𝜓𝜒) → (𝜂𝜁)))
1211pm2.43i 50 . . . . 5 ((𝜑𝜓𝜒) → (𝜂𝜁))
133, 12syl5com 31 . . . 4 ((𝜑𝜓𝜒) → ((𝜑𝜓𝜒) → 𝜁))
1413pm2.43i 50 . . 3 ((𝜑𝜓𝜒) → 𝜁)
15143exp 1256 . 2 (𝜑 → (𝜓 → (𝜒𝜁)))
1615dfvd3ir 37830 1 (   𝜑   ,   𝜓   ,   𝜒   ▶   𝜁   )
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ w3a 1031  (   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-vd3 37827 This theorem is referenced by:  e33  37982  e123  38010
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