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Theorem un0.1 33762
Description: T. is the constant true, a tautology (see df-tru 1398). 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 33534 . . 3  |-  ( T. 
->  ph )
3 un0.1.2 . . . 4  |-  (. ps  ->.  ch
).
43in1 33534 . . 3  |-  ( ps 
->  ch )
5 un0.1.3 . . . 4  |-  (. (. T.  ,. ps ).  ->.  th ).
65dfvd2ani 33546 . . 3  |-  ( ( T.  /\  ps )  ->  th )
72, 4, 6uun0.1 33761 . 2  |-  ( ps 
->  th )
87dfvd1ir 33536 1  |-  (. ps  ->.  th
).
Colors of variables: wff setvar class
Syntax hints:   T. wtru 1396   (.wvd1 33532   (.wvhc2 33543
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 185  df-an 371  df-tru 1398  df-vd1 33533  df-vhc2 33544
This theorem is referenced by:  sspwimpVD  33905
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