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Theorem un0.1 28234
Description:  T. is the constant true, a tautology ( see: df-tru 1325). 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 28005 . . 3  |-  (  T. 
->  ph )
3 un0.1.2 . . . 4  |-  (. ps  ->.  ch
).
43in1 28005 . . 3  |-  ( ps 
->  ch )
5 un0.1.3 . . . 4  |-  (. (.  T.  ,. ps ).  ->.  th ).
65dfvd2ani 28018 . . 3  |-  ( (  T.  /\  ps )  ->  th )
72, 4, 6uun0.1 28233 . 2  |-  ( ps 
->  th )
87dfvd1ir 28007 1  |-  (. ps  ->.  th
).
Colors of variables: wff set class
Syntax hints:    T. wtru 1322   (.wvd1 28003   (.wvhc2 28015
This theorem is referenced by:  sspwimpVD  28374
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-vd1 28004  df-vhc2 28016
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