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Theorem re1luk3 1467
Description: luk-3 1412 derived from the Tarski-Bernays-Wajsberg axioms.

This theorem, along with re1luk1 1465 and re1luk2 1466 proves that tbw-ax1 1455, tbw-ax2 1456, tbw-ax3 1457, and tbw-ax4 1458, with ax-mp 8 can be used as a complete axiom system for all of propositional calculus. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion
Ref Expression
re1luk3  |-  ( ph  ->  ( -.  ph  ->  ps ) )

Proof of Theorem re1luk3
StepHypRef Expression
1 tbw-negdf 1454 . . 3  |-  ( ( ( -.  ph  ->  (
ph  ->  F.  ) )  ->  ( ( ( ph  ->  F.  )  ->  -.  ph )  ->  F.  )
)  ->  F.  )
2 tbwlem5 1464 . . 3  |-  ( ( ( ( -.  ph  ->  ( ph  ->  F.  ) )  ->  (
( ( ph  ->  F.  )  ->  -.  ph )  ->  F.  ) )  ->  F.  )  ->  ( -. 
ph  ->  ( ph  ->  F.  ) ) )
31, 2ax-mp 8 . 2  |-  ( -. 
ph  ->  ( ph  ->  F.  ) )
4 tbw-ax4 1458 . . . 4  |-  (  F. 
->  ps )
5 tbw-ax1 1455 . . . . 5  |-  ( (
ph  ->  F.  )  ->  ( (  F.  ->  ps )  ->  ( ph  ->  ps ) ) )
6 tbwlem1 1460 . . . . 5  |-  ( ( ( ph  ->  F.  )  ->  ( (  F. 
->  ps )  ->  ( ph  ->  ps ) ) )  ->  ( (  F.  ->  ps )  -> 
( ( ph  ->  F.  )  ->  ( ph  ->  ps ) ) ) )
75, 6ax-mp 8 . . . 4  |-  ( (  F.  ->  ps )  ->  ( ( ph  ->  F.  )  ->  ( ph  ->  ps ) ) )
84, 7ax-mp 8 . . 3  |-  ( (
ph  ->  F.  )  ->  (
ph  ->  ps ) )
9 tbwlem1 1460 . . 3  |-  ( ( ( ph  ->  F.  )  ->  ( ph  ->  ps ) )  ->  ( ph  ->  ( ( ph  ->  F.  )  ->  ps ) ) )
108, 9ax-mp 8 . 2  |-  ( ph  ->  ( ( ph  ->  F.  )  ->  ps )
)
11 tbw-ax1 1455 . 2  |-  ( ( -.  ph  ->  ( ph  ->  F.  ) )  -> 
( ( ( ph  ->  F.  )  ->  ps )  ->  ( -.  ph  ->  ps ) ) )
123, 10, 11mpsyl 59 1  |-  ( ph  ->  ( -.  ph  ->  ps ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    F. wfal 1308
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 177  df-tru 1310  df-fal 1311
  Copyright terms: Public domain W3C validator