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Theorem e111 41163
Description: A virtual deduction elimination rule (see syl3c 66). (Contributed by Alan Sare, 14-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
e111.1 (   𝜑   ▶   𝜓   )
e111.2 (   𝜑   ▶   𝜒   )
e111.3 (   𝜑   ▶   𝜃   )
e111.4 (𝜓 → (𝜒 → (𝜃𝜏)))
Assertion
Ref Expression
e111 (   𝜑   ▶   𝜏   )

Proof of Theorem e111
StepHypRef Expression
1 e111.3 . . . . 5 (   𝜑   ▶   𝜃   )
21in1 41060 . . . 4 (𝜑𝜃)
3 e111.1 . . . . . . 7 (   𝜑   ▶   𝜓   )
43in1 41060 . . . . . 6 (𝜑𝜓)
5 e111.2 . . . . . . 7 (   𝜑   ▶   𝜒   )
65in1 41060 . . . . . 6 (𝜑𝜒)
7 e111.4 . . . . . 6 (𝜓 → (𝜒 → (𝜃𝜏)))
84, 6, 7syl2im 40 . . . . 5 (𝜑 → (𝜑 → (𝜃𝜏)))
98pm2.43i 52 . . . 4 (𝜑 → (𝜃𝜏))
102, 9syl5com 31 . . 3 (𝜑 → (𝜑𝜏))
1110pm2.43i 52 . 2 (𝜑𝜏)
1211dfvd1ir 41062 1 (   𝜑   ▶   𝜏   )
Colors of variables: wff setvar class
Syntax hints:  wi 4  (   wvd1 41058
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 209  df-vd1 41059
This theorem is referenced by:  e110  41165  e101  41167  e011  41169  e100  41171  e010  41173  e001  41175  e11  41177  ordelordALTVD  41356
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