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Theorem e223 39177
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 39147 . . . 4 (   𝜑   ▶   (𝜓𝜒)   )
32in1 39104 . . 3 (𝜑 → (𝜓𝜒))
4 e223.2 . . . . 5 (   𝜑   ,   𝜓   ▶   𝜃   )
54in2 39147 . . . 4 (   𝜑   ▶   (𝜓𝜃)   )
65in1 39104 . . 3 (𝜑 → (𝜓𝜃))
7 e223.3 . . . . . 6 (   𝜑   ,   𝜓   ,   𝜏   ▶   𝜂   )
87in3 39151 . . . . 5 (   𝜑   ,   𝜓   ▶   (𝜏𝜂)   )
98in2 39147 . . . 4 (   𝜑   ▶   (𝜓 → (𝜏𝜂))   )
109in1 39104 . . 3 (𝜑 → (𝜓 → (𝜏𝜂)))
11 e223.4 . . 3 (𝜒 → (𝜃 → (𝜂𝜁)))
123, 6, 10, 11ee223 39176 . 2 (𝜑 → (𝜓 → (𝜏𝜁)))
1312dfvd3ir 39126 1 (   𝜑   ,   𝜓   ,   𝜏   ▶   𝜁   )
Colors of variables: wff setvar class
Syntax hints:  wi 4  (   wvd2 39110  (   wvd3 39120
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-3an 1056  df-vd1 39103  df-vd2 39111  df-vd3 39123
This theorem is referenced by:  tratrbVD  39411
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