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Theorem re2luk3 1522
Description: luk-3 1412 derived from Russell-Bernays'.

This theorem, along with re1axmp 1519, re2luk1 1520, and re2luk2 1521 shows that rb-ax1 1507, rb-ax2 1508, rb-ax3 1509, and rb-ax4 1510, along with anmp 1506, can be used as a complete axiomatization of propositional calculus. (Contributed by Anthony Hart, 19-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion
Ref Expression
re2luk3  |-  ( ph  ->  ( -.  ph  ->  ps ) )

Proof of Theorem re2luk3
StepHypRef Expression
1 rb-imdf 1505 . . . 4  |-  -.  ( -.  ( -.  ( -. 
ph  ->  ps )  \/  ( -.  -.  ph  \/  ps ) )  \/ 
-.  ( -.  ( -.  -.  ph  \/  ps )  \/  ( -.  ph 
->  ps ) ) )
21rblem7 1518 . . 3  |-  ( -.  ( -.  -.  ph  \/  ps )  \/  ( -.  ph  ->  ps )
)
3 rb-ax4 1510 . . . . . 6  |-  ( -.  ( -.  ph  \/  -.  ph )  \/  -.  ph )
4 rb-ax3 1509 . . . . . 6  |-  ( -. 
-.  ph  \/  ( -.  ph  \/  -.  ph ) )
53, 4rbsyl 1511 . . . . 5  |-  ( -. 
-.  ph  \/  -.  ph )
6 rb-ax2 1508 . . . . 5  |-  ( -.  ( -.  -.  ph  \/  -.  ph )  \/  ( -.  ph  \/  -.  -.  ph ) )
75, 6anmp 1506 . . . 4  |-  ( -. 
ph  \/  -.  -.  ph )
8 rblem2 1513 . . . 4  |-  ( -.  ( -.  ph  \/  -.  -.  ph )  \/  ( -.  ph  \/  ( -.  -.  ph  \/  ps ) ) )
97, 8anmp 1506 . . 3  |-  ( -. 
ph  \/  ( -.  -.  ph  \/  ps )
)
102, 9rbsyl 1511 . 2  |-  ( -. 
ph  \/  ( -.  ph 
->  ps ) )
11 rb-imdf 1505 . . 3  |-  -.  ( -.  ( -.  ( ph  ->  ( -.  ph  ->  ps ) )  \/  ( -.  ph  \/  ( -. 
ph  ->  ps ) ) )  \/  -.  ( -.  ( -.  ph  \/  ( -.  ph  ->  ps ) )  \/  ( ph  ->  ( -.  ph  ->  ps ) ) ) )
1211rblem7 1518 . 2  |-  ( -.  ( -.  ph  \/  ( -.  ph  ->  ps ) )  \/  ( ph  ->  ( -.  ph  ->  ps ) ) )
1310, 12anmp 1506 1  |-  ( ph  ->  ( -.  ph  ->  ps ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 357
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-or 359  df-an 360
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