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Theorem el123 41457
 Description: A virtual deduction elimination rule. (Contributed by Alan Sare, 13-Jun-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
el123.1 (   𝜑   ▶   𝜓   )
el123.2 (   𝜒   ▶   𝜃   )
el123.3 (   𝜏   ▶   𝜂   )
el123.4 ((𝜓𝜃𝜂) → 𝜁)
Assertion
Ref Expression
el123 (   (   𝜑   ,   𝜒   ,   𝜏   )   ▶   𝜁   )

Proof of Theorem el123
StepHypRef Expression
1 el123.1 . . . 4 (   𝜑   ▶   𝜓   )
21in1 41264 . . 3 (𝜑𝜓)
3 el123.2 . . . 4 (   𝜒   ▶   𝜃   )
43in1 41264 . . 3 (𝜒𝜃)
5 el123.3 . . . 4 (   𝜏   ▶   𝜂   )
65in1 41264 . . 3 (𝜏𝜂)
7 el123.4 . . 3 ((𝜓𝜃𝜂) → 𝜁)
82, 4, 6, 7syl3an 1157 . 2 ((𝜑𝜒𝜏) → 𝜁)
98dfvd3anir 41289 1 (   (   𝜑   ,   𝜒   ,   𝜏   )   ▶   𝜁   )
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ w3a 1084  (   wvd1 41262  (   wvhc3 41281 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 210  df-an 400  df-3an 1086  df-vd1 41263  df-vhc3 41282 This theorem is referenced by:  suctrALTcfVD  41616
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