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Theorem un0.1 28818
Description:  T. is the constant true, a tautology ( see: df-tru 1328). 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 28589 . . 3  |-  (  T. 
->  ph )
3 un0.1.2 . . . 4  |-  (. ps  ->.  ch
).
43in1 28589 . . 3  |-  ( ps 
->  ch )
5 un0.1.3 . . . 4  |-  (. (.  T.  ,. ps ).  ->.  th ).
65dfvd2ani 28602 . . 3  |-  ( (  T.  /\  ps )  ->  th )
72, 4, 6uun0.1 28817 . 2  |-  ( ps 
->  th )
87dfvd1ir 28591 1  |-  (. ps  ->.  th
).
Colors of variables: wff set class
Syntax hints:    T. wtru 1325   (.wvd1 28587   (.wvhc2 28599
This theorem is referenced by:  sspwimpVD  28958
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 1328  df-vd1 28588  df-vhc2 28600
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