MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  re1tbw4 Structured version   Unicode version

Theorem re1tbw4 1625
Description: tbw-ax4 1580 rederived from merco2 1613.

This theorem, along with re1tbw1 1622, re1tbw2 1623, and re1tbw3 1624, shows that merco2 1613, along with ax-mp 5, 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 1624 . . 3  |-  ( ( ( ph  ->  ph )  ->  ph )  ->  ph )
2 re1tbw2 1623 . . . 4  |-  ( ph  ->  ( ( ph  ->  ph )  ->  ph ) )
3 re1tbw1 1622 . . . 4  |-  ( (
ph  ->  ( ( ph  ->  ph )  ->  ph )
)  ->  ( (
( ( ph  ->  ph )  ->  ph )  ->  ph )  ->  ( ph  ->  ph ) ) )
42, 3ax-mp 5 . . 3  |-  ( ( ( ( ph  ->  ph )  ->  ph )  ->  ph )  ->  ( ph  ->  ph ) )
51, 4ax-mp 5 . 2  |-  ( ph  ->  ph )
6 re1tbw3 1624 . . . . 5  |-  ( ( ( ( F.  ->  ph )  ->  ph )  -> 
( F.  ->  ph )
)  ->  ( F.  ->  ph ) )
7 re1tbw2 1623 . . . . . 6  |-  ( ( F.  ->  ph )  -> 
( ( ( F. 
->  ph )  ->  ph )  ->  ( F.  ->  ph )
) )
8 re1tbw1 1622 . . . . . 6  |-  ( ( ( F.  ->  ph )  ->  ( ( ( F. 
->  ph )  ->  ph )  ->  ( F.  ->  ph )
) )  ->  (
( ( ( ( F.  ->  ph )  ->  ph )  ->  ( F. 
->  ph ) )  -> 
( F.  ->  ph )
)  ->  ( ( F.  ->  ph )  ->  ( F.  ->  ph ) ) ) )
97, 8ax-mp 5 . . . . 5  |-  ( ( ( ( ( F. 
->  ph )  ->  ph )  ->  ( F.  ->  ph )
)  ->  ( F.  ->  ph ) )  -> 
( ( F.  ->  ph )  ->  ( F.  ->  ph ) ) )
106, 9ax-mp 5 . . . 4  |-  ( ( F.  ->  ph )  -> 
( F.  ->  ph )
)
11 mercolem3 1616 . . . . 5  |-  ( ( ( F.  ->  ph )  ->  ph )  ->  (
( F.  ->  ph )  ->  ( F.  ->  ph )
) )
12 merco2 1613 . . . . 5  |-  ( ( ( ( F.  ->  ph )  ->  ph )  -> 
( ( F.  ->  ph )  ->  ( F.  ->  ph ) ) )  ->  ( ( ( F.  ->  ph )  -> 
( F.  ->  ph )
)  ->  ( ( ph  ->  ph )  ->  (
( ph  ->  ph )  ->  ( F.  ->  ph )
) ) ) )
1311, 12ax-mp 5 . . . 4  |-  ( ( ( F.  ->  ph )  ->  ( F.  ->  ph )
)  ->  ( ( ph  ->  ph )  ->  (
( ph  ->  ph )  ->  ( F.  ->  ph )
) ) )
1410, 13ax-mp 5 . . 3  |-  ( (
ph  ->  ph )  ->  (
( ph  ->  ph )  ->  ( F.  ->  ph )
) )
155, 14ax-mp 5 . 2  |-  ( (
ph  ->  ph )  ->  ( F.  ->  ph ) )
165, 15ax-mp 5 1  |-  ( F. 
->  ph )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   F. wfal 1442
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 188  df-tru 1440  df-fal 1443
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator