MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  merco1lem14 Unicode version

Theorem merco1lem14 1485
Description: Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1468. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
merco1lem14  |-  ( ( ( ( ph  ->  ps )  ->  ps )  ->  ch )  ->  ( ph  ->  ch ) )

Proof of Theorem merco1lem14
StepHypRef Expression
1 merco1lem13 1484 . . . 4  |-  ( ( ( ( ph  ->  ps )  ->  ( ( ph  ->  ps )  ->  ps ) )  ->  (
( ph  ->  ps )  ->  ps ) )  -> 
( ph  ->  ( (
ph  ->  ps )  ->  ps ) ) )
2 merco1lem8 1479 . . . . . 6  |-  ( ( ( ( ( ph  ->  ( ( ph  ->  ps )  ->  ps )
)  ->  ph )  -> 
( ( ( ( ( ph  ->  ps )  ->  ( ( ph  ->  ps )  ->  ps ) )  ->  (
( ph  ->  ps )  ->  ps ) )  -> 
( ph  ->  ( (
ph  ->  ps )  ->  ps ) ) )  ->  F.  ) )  ->  ph )  ->  ( ( ( ph  ->  ps )  ->  (
( ph  ->  ps )  ->  ps ) )  -> 
( ( ph  ->  ps )  ->  ps )
) )
3 merco1 1468 . . . . . 6  |-  ( ( ( ( ( (
ph  ->  ( ( ph  ->  ps )  ->  ps ) )  ->  ph )  ->  ( ( ( ( ( ph  ->  ps )  ->  ( ( ph  ->  ps )  ->  ps ) )  ->  (
( ph  ->  ps )  ->  ps ) )  -> 
( ph  ->  ( (
ph  ->  ps )  ->  ps ) ) )  ->  F.  ) )  ->  ph )  ->  ( ( ( ph  ->  ps )  ->  (
( ph  ->  ps )  ->  ps ) )  -> 
( ( ph  ->  ps )  ->  ps )
) )  ->  (
( ( ( (
ph  ->  ps )  -> 
( ( ph  ->  ps )  ->  ps )
)  ->  ( ( ph  ->  ps )  ->  ps ) )  ->  ( ph  ->  ( ( ph  ->  ps )  ->  ps ) ) )  -> 
( ( ( ( ( ph  ->  ps )  ->  ( ( ph  ->  ps )  ->  ps ) )  ->  (
( ph  ->  ps )  ->  ps ) )  -> 
( ph  ->  ( (
ph  ->  ps )  ->  ps ) ) )  -> 
( ph  ->  ( (
ph  ->  ps )  ->  ps ) ) ) ) )
42, 3ax-mp 8 . . . . 5  |-  ( ( ( ( ( ph  ->  ps )  ->  (
( ph  ->  ps )  ->  ps ) )  -> 
( ( ph  ->  ps )  ->  ps )
)  ->  ( ph  ->  ( ( ph  ->  ps )  ->  ps )
) )  ->  (
( ( ( (
ph  ->  ps )  -> 
( ( ph  ->  ps )  ->  ps )
)  ->  ( ( ph  ->  ps )  ->  ps ) )  ->  ( ph  ->  ( ( ph  ->  ps )  ->  ps ) ) )  -> 
( ph  ->  ( (
ph  ->  ps )  ->  ps ) ) ) )
5 merco1lem9 1480 . . . . 5  |-  ( ( ( ( ( (
ph  ->  ps )  -> 
( ( ph  ->  ps )  ->  ps )
)  ->  ( ( ph  ->  ps )  ->  ps ) )  ->  ( ph  ->  ( ( ph  ->  ps )  ->  ps ) ) )  -> 
( ( ( ( ( ph  ->  ps )  ->  ( ( ph  ->  ps )  ->  ps ) )  ->  (
( ph  ->  ps )  ->  ps ) )  -> 
( ph  ->  ( (
ph  ->  ps )  ->  ps ) ) )  -> 
( ph  ->  ( (
ph  ->  ps )  ->  ps ) ) ) )  ->  ( ( ( ( ( ph  ->  ps )  ->  ( ( ph  ->  ps )  ->  ps ) )  ->  (
( ph  ->  ps )  ->  ps ) )  -> 
( ph  ->  ( (
ph  ->  ps )  ->  ps ) ) )  -> 
( ph  ->  ( (
ph  ->  ps )  ->  ps ) ) ) )
64, 5ax-mp 8 . . . 4  |-  ( ( ( ( ( ph  ->  ps )  ->  (
( ph  ->  ps )  ->  ps ) )  -> 
( ( ph  ->  ps )  ->  ps )
)  ->  ( ph  ->  ( ( ph  ->  ps )  ->  ps )
) )  ->  ( ph  ->  ( ( ph  ->  ps )  ->  ps ) ) )
71, 6ax-mp 8 . . 3  |-  ( ph  ->  ( ( ph  ->  ps )  ->  ps )
)
8 merco1lem12 1483 . . 3  |-  ( (
ph  ->  ( ( ph  ->  ps )  ->  ps ) )  ->  (
( ( ( ch 
->  ph )  ->  ( ph  ->  F.  ) )  ->  ph )  ->  (
( ph  ->  ps )  ->  ps ) ) )
97, 8ax-mp 8 . 2  |-  ( ( ( ( ch  ->  ph )  ->  ( ph  ->  F.  ) )  ->  ph )  ->  ( (
ph  ->  ps )  ->  ps ) )
10 merco1 1468 . 2  |-  ( ( ( ( ( ch 
->  ph )  ->  ( ph  ->  F.  ) )  ->  ph )  ->  (
( ph  ->  ps )  ->  ps ) )  -> 
( ( ( (
ph  ->  ps )  ->  ps )  ->  ch )  ->  ( ph  ->  ch ) ) )
119, 10ax-mp 8 1  |-  ( ( ( ( ph  ->  ps )  ->  ps )  ->  ch )  ->  ( ph  ->  ch ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    F. wfal 1308
This theorem is referenced by:  merco1lem15  1486  retbwax1  1490
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