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Theorem looinvdc 885
Description: The Inversion Axiom of the infinite-valued sentential logic (L-infinity) of Lukasiewicz, but where one of the propositions is decidable. Using dfor2dc 865, we can see that this expresses "disjunction commutes." Theorem *2.69 of [WhiteheadRussell] p. 108 (plus the decidability condition). (Contributed by NM, 12-Aug-2004.)
Assertion
Ref Expression
looinvdc  |-  (DECID  ph  ->  ( ( ( ph  ->  ps )  ->  ps )  ->  ( ( ps  ->  ph )  ->  ph ) ) )

Proof of Theorem looinvdc
StepHypRef Expression
1 imim1 76 . 2  |-  ( ( ( ph  ->  ps )  ->  ps )  -> 
( ( ps  ->  ph )  ->  ( ( ph  ->  ps )  ->  ph ) ) )
2 peircedc 884 . 2  |-  (DECID  ph  ->  ( ( ( ph  ->  ps )  ->  ph )  ->  ph ) )
31, 2syl9r 73 1  |-  (DECID  ph  ->  ( ( ( ph  ->  ps )  ->  ps )  ->  ( ( ps  ->  ph )  ->  ph ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4  DECID wdc 804
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in2 589  ax-io 683
This theorem depends on definitions:  df-bi 116  df-dc 805
This theorem is referenced by: (None)
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