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Theorem un0.1 39506
Description: is the constant true, a tautology (see df-tru 1633). Kleene's "empty conjunction" is logically equivalent to . In a virtual deduction we shall interpret to be the empty wff or the empty collection of virtual hypotheses. in a virtual deduction translated into conventional notation we shall interpret to be Kleene's empty conjunction. If 𝜃 is true given the empty collection of virtual hypotheses and another collection of virtual hypotheses, then it is true given only the other collection of virtual hypotheses. (Contributed by Alan Sare, 23-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
un0.1.1 (      ▶   𝜑   )
un0.1.2 (   𝜓   ▶   𝜒   )
un0.1.3 (   (      ,   𝜓   )   ▶   𝜃   )
Assertion
Ref Expression
un0.1 (   𝜓   ▶   𝜃   )

Proof of Theorem un0.1
StepHypRef Expression
1 un0.1.1 . . . 4 (      ▶   𝜑   )
21in1 39287 . . 3 (⊤ → 𝜑)
3 un0.1.2 . . . 4 (   𝜓   ▶   𝜒   )
43in1 39287 . . 3 (𝜓𝜒)
5 un0.1.3 . . . 4 (   (      ,   𝜓   )   ▶   𝜃   )
65dfvd2ani 39299 . . 3 ((⊤ ∧ 𝜓) → 𝜃)
72, 4, 6uun0.1 39505 . 2 (𝜓𝜃)
87dfvd1ir 39289 1 (   𝜓   ▶   𝜃   )
Colors of variables: wff setvar class
Syntax hints:  wtru 1631  (   wvd1 39285  (   wvhc2 39296
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-tru 1633  df-vd1 39286  df-vhc2 39297
This theorem is referenced by:  sspwimpVD  39652
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