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Theorem re1tbw4 1503
Description: tbw-ax4 1458 rederived from merco2 1491.

This theorem, along with re1tbw1 1500, re1tbw2 1501, and re1tbw3 1502, shows that merco2 1491, along with ax-mp 8, can be used as a complete axiomatization of propositional calculus. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion
Ref Expression
re1tbw4  |-  (  F. 
->  ph )

Proof of Theorem re1tbw4
StepHypRef Expression
1 re1tbw3 1502 . . 3  |-  ( ( ( ph  ->  ph )  ->  ph )  ->  ph )
2 re1tbw2 1501 . . . 4  |-  ( ph  ->  ( ( ph  ->  ph )  ->  ph ) )
3 re1tbw1 1500 . . . 4  |-  ( (
ph  ->  ( ( ph  ->  ph )  ->  ph )
)  ->  ( (
( ( ph  ->  ph )  ->  ph )  ->  ph )  ->  ( ph  ->  ph ) ) )
42, 3ax-mp 8 . . 3  |-  ( ( ( ( ph  ->  ph )  ->  ph )  ->  ph )  ->  ( ph  ->  ph ) )
51, 4ax-mp 8 . 2  |-  ( ph  ->  ph )
6 re1tbw3 1502 . . . . 5  |-  ( ( ( (  F.  ->  ph )  ->  ph )  -> 
(  F.  ->  ph )
)  ->  (  F.  ->  ph ) )
7 re1tbw2 1501 . . . . . 6  |-  ( (  F.  ->  ph )  -> 
( ( (  F. 
->  ph )  ->  ph )  ->  (  F.  ->  ph )
) )
8 re1tbw1 1500 . . . . . 6  |-  ( ( (  F.  ->  ph )  ->  ( ( (  F. 
->  ph )  ->  ph )  ->  (  F.  ->  ph )
) )  ->  (
( ( ( (  F.  ->  ph )  ->  ph )  ->  (  F. 
->  ph ) )  -> 
(  F.  ->  ph )
)  ->  ( (  F.  ->  ph )  ->  (  F.  ->  ph ) ) ) )
97, 8ax-mp 8 . . . . 5  |-  ( ( ( ( (  F. 
->  ph )  ->  ph )  ->  (  F.  ->  ph )
)  ->  (  F.  ->  ph ) )  -> 
( (  F.  ->  ph )  ->  (  F.  ->  ph ) ) )
106, 9ax-mp 8 . . . 4  |-  ( (  F.  ->  ph )  -> 
(  F.  ->  ph )
)
11 mercolem3 1494 . . . . 5  |-  ( ( (  F.  ->  ph )  ->  ph )  ->  (
(  F.  ->  ph )  ->  (  F.  ->  ph )
) )
12 merco2 1491 . . . . 5  |-  ( ( ( (  F.  ->  ph )  ->  ph )  -> 
( (  F.  ->  ph )  ->  (  F.  ->  ph ) ) )  ->  ( ( (  F.  ->  ph )  -> 
(  F.  ->  ph )
)  ->  ( ( ph  ->  ph )  ->  (
( ph  ->  ph )  ->  (  F.  ->  ph )
) ) ) )
1311, 12ax-mp 8 . . . 4  |-  ( ( (  F.  ->  ph )  ->  (  F.  ->  ph )
)  ->  ( ( ph  ->  ph )  ->  (
( ph  ->  ph )  ->  (  F.  ->  ph )
) ) )
1410, 13ax-mp 8 . . 3  |-  ( (
ph  ->  ph )  ->  (
( ph  ->  ph )  ->  (  F.  ->  ph )
) )
155, 14ax-mp 8 . 2  |-  ( (
ph  ->  ph )  ->  (  F.  ->  ph ) )
165, 15ax-mp 8 1  |-  (  F. 
->  ph )
Colors of variables: wff set class
Syntax hints:    -> 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