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Theorem mercolem7 1517
Description: Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 1510. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
mercolem7  |-  ( (
ph  ->  ps )  -> 
( ( ( ph  ->  ch )  ->  ( th  ->  ps ) )  ->  ( th  ->  ps ) ) )

Proof of Theorem mercolem7
StepHypRef Expression
1 merco2 1510 . 2  |-  ( ( ( ph  ->  ph )  ->  ( (  F.  ->  ph )  ->  ph ) )  ->  ( ( ph  ->  ph )  ->  ( ph  ->  ( ph  ->  ph ) ) ) )
2 mercolem3 1513 . . . 4  |-  ( ( ( ph  ->  ch )  ->  ( th  ->  ps ) )  ->  (
( ph  ->  ch )  ->  ( ( ( ph  ->  ch )  ->  ( th  ->  ps ) )  ->  ( th  ->  ps ) ) ) )
3 mercolem6 1516 . . . 4  |-  ( ( ( ( ph  ->  ch )  ->  ( th  ->  ps ) )  -> 
( ( ph  ->  ch )  ->  ( (
( ph  ->  ch )  ->  ( th  ->  ps ) )  ->  ( th  ->  ps ) ) ) )  ->  (
( ph  ->  ch )  ->  ( ( ( ph  ->  ch )  ->  ( th  ->  ps ) )  ->  ( th  ->  ps ) ) ) )
42, 3ax-mp 8 . . 3  |-  ( (
ph  ->  ch )  -> 
( ( ( ph  ->  ch )  ->  ( th  ->  ps ) )  ->  ( th  ->  ps ) ) )
5 mercolem5 1515 . . . 4  |-  ( ph  ->  ( ( ph  ->  ps )  ->  ( (
( ph  ->  ch )  ->  ( th  ->  ps ) )  ->  ( th  ->  ps ) ) ) )
6 mercolem4 1514 . . . 4  |-  ( (
ph  ->  ( ( ph  ->  ps )  ->  (
( ( ph  ->  ch )  ->  ( th  ->  ps ) )  -> 
( th  ->  ps ) ) ) )  ->  ( ( (
ph  ->  ch )  -> 
( ( ( ph  ->  ch )  ->  ( th  ->  ps ) )  ->  ( th  ->  ps ) ) )  -> 
( ( ( (
ph  ->  ph )  ->  (
(  F.  ->  ph )  ->  ph ) )  -> 
( ( ph  ->  ph )  ->  ( ph  ->  ( ph  ->  ph )
) ) )  -> 
( ( ph  ->  ps )  ->  ( (
( ph  ->  ch )  ->  ( th  ->  ps ) )  ->  ( th  ->  ps ) ) ) ) ) )
75, 6ax-mp 8 . . 3  |-  ( ( ( ph  ->  ch )  ->  ( ( (
ph  ->  ch )  -> 
( th  ->  ps ) )  ->  ( th  ->  ps ) ) )  ->  ( (
( ( ph  ->  ph )  ->  ( (  F.  ->  ph )  ->  ph )
)  ->  ( ( ph  ->  ph )  ->  ( ph  ->  ( ph  ->  ph ) ) ) )  ->  ( ( ph  ->  ps )  ->  (
( ( ph  ->  ch )  ->  ( th  ->  ps ) )  -> 
( th  ->  ps ) ) ) ) )
84, 7ax-mp 8 . 2  |-  ( ( ( ( ph  ->  ph )  ->  ( (  F.  ->  ph )  ->  ph )
)  ->  ( ( ph  ->  ph )  ->  ( ph  ->  ( ph  ->  ph ) ) ) )  ->  ( ( ph  ->  ps )  ->  (
( ( ph  ->  ch )  ->  ( th  ->  ps ) )  -> 
( th  ->  ps ) ) ) )
91, 8ax-mp 8 1  |-  ( (
ph  ->  ps )  -> 
( ( ( ph  ->  ch )  ->  ( th  ->  ps ) )  ->  ( th  ->  ps ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    F. wfal 1326
This theorem is referenced by:  mercolem8  1518
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 178  df-tru 1328  df-fal 1329
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