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Theorem un0.1 37166
Description: T. is the constant true, a tautology (see df-tru 1447). Kleene's "empty conjunction" is logically equivalent to T.. In a virtual deduction we shall interpret T. to be the empty wff or the empty collection of virtual hypotheses. T. in a virtual deduction translated into conventional notation we shall interpret to be Kleene's empty conjunction. If  th 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  |-  (. T.  ->.  ph ).
un0.1.2  |-  (. ps  ->.  ch
).
un0.1.3  |-  (. (. T.  ,. ps ).  ->.  th ).
Assertion
Ref Expression
un0.1  |-  (. ps  ->.  th
).

Proof of Theorem un0.1
StepHypRef Expression
1 un0.1.1 . . . 4  |-  (. T.  ->.  ph ).
21in1 36941 . . 3  |-  ( T. 
->  ph )
3 un0.1.2 . . . 4  |-  (. ps  ->.  ch
).
43in1 36941 . . 3  |-  ( ps 
->  ch )
5 un0.1.3 . . . 4  |-  (. (. T.  ,. ps ).  ->.  th ).
65dfvd2ani 36953 . . 3  |-  ( ( T.  /\  ps )  ->  th )
72, 4, 6uun0.1 37165 . 2  |-  ( ps 
->  th )
87dfvd1ir 36943 1  |-  (. ps  ->.  th
).
Colors of variables: wff setvar class
Syntax hints:   T. wtru 1445   (.wvd1 36939   (.wvhc2 36950
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 189  df-an 373  df-tru 1447  df-vd1 36940  df-vhc2 36951
This theorem is referenced by:  sspwimpVD  37316
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