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Theorem mercolem5 1496
Description: Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 1491. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
mercolem5  |-  ( th 
->  ( ( th  ->  ph )  ->  ( ta  ->  ( ch  ->  ph )
) ) )

Proof of Theorem mercolem5
StepHypRef Expression
1 merco2 1491 . 2  |-  ( ( ( ph  ->  ph )  ->  ( (  F.  ->  ph )  ->  ph ) )  ->  ( ( ph  ->  ph )  ->  ( ph  ->  ( ph  ->  ph ) ) ) )
2 merco2 1491 . . . . 5  |-  ( ( ( ph  ->  ph )  ->  ( (  F.  ->  ph )  ->  th )
)  ->  ( ( th  ->  ph )  ->  ( ta  ->  ( ch  ->  ph ) ) ) )
3 mercolem1 1492 . . . . 5  |-  ( ( ( ( ph  ->  ph )  ->  ( (  F.  ->  ph )  ->  th )
)  ->  ( ( th  ->  ph )  ->  ( ta  ->  ( ch  ->  ph ) ) ) )  ->  ( ( (  F.  ->  ph )  ->  th )  ->  ( th 
->  ( ( th  ->  ph )  ->  ( ta  ->  ( ch  ->  ph )
) ) ) ) )
42, 3ax-mp 8 . . . 4  |-  ( ( (  F.  ->  ph )  ->  th )  ->  ( th  ->  ( ( th 
->  ph )  ->  ( ta  ->  ( ch  ->  ph ) ) ) ) )
5 mercolem2 1493 . . . . 5  |-  ( ( ( th  ->  (
( th  ->  ph )  ->  ( ta  ->  ( ch  ->  ph ) ) ) )  ->  th )  ->  ( (  F.  ->  ph )  ->  ( (  F.  ->  ph )  ->  th )
) )
6 merco2 1491 . . . . 5  |-  ( ( ( ( th  ->  ( ( th  ->  ph )  ->  ( ta  ->  ( ch  ->  ph ) ) ) )  ->  th )  ->  ( (  F.  ->  ph )  ->  ( (  F.  ->  ph )  ->  th )
) )  ->  (
( ( (  F. 
->  ph )  ->  th )  ->  ( th  ->  (
( th  ->  ph )  ->  ( ta  ->  ( ch  ->  ph ) ) ) ) )  ->  (
( ( ( ph  ->  ph )  ->  (
(  F.  ->  ph )  ->  ph ) )  -> 
( ( ph  ->  ph )  ->  ( ph  ->  ( ph  ->  ph )
) ) )  -> 
( ( ( (
ph  ->  ph )  ->  (
(  F.  ->  ph )  ->  ph ) )  -> 
( ( ph  ->  ph )  ->  ( ph  ->  ( ph  ->  ph )
) ) )  -> 
( th  ->  (
( th  ->  ph )  ->  ( ta  ->  ( ch  ->  ph ) ) ) ) ) ) ) )
75, 6ax-mp 8 . . . 4  |-  ( ( ( (  F.  ->  ph )  ->  th )  ->  ( th  ->  (
( th  ->  ph )  ->  ( ta  ->  ( ch  ->  ph ) ) ) ) )  ->  (
( ( ( ph  ->  ph )  ->  (
(  F.  ->  ph )  ->  ph ) )  -> 
( ( ph  ->  ph )  ->  ( ph  ->  ( ph  ->  ph )
) ) )  -> 
( ( ( (
ph  ->  ph )  ->  (
(  F.  ->  ph )  ->  ph ) )  -> 
( ( ph  ->  ph )  ->  ( ph  ->  ( ph  ->  ph )
) ) )  -> 
( th  ->  (
( th  ->  ph )  ->  ( ta  ->  ( ch  ->  ph ) ) ) ) ) ) )
84, 7ax-mp 8 . . 3  |-  ( ( ( ( ph  ->  ph )  ->  ( (  F.  ->  ph )  ->  ph )
)  ->  ( ( ph  ->  ph )  ->  ( ph  ->  ( ph  ->  ph ) ) ) )  ->  ( ( ( ( ph  ->  ph )  ->  ( (  F.  ->  ph )  ->  ph ) )  ->  ( ( ph  ->  ph )  ->  ( ph  ->  ( ph  ->  ph ) ) ) )  ->  ( th  ->  ( ( th  ->  ph )  ->  ( ta  ->  ( ch  ->  ph ) ) ) ) ) )
91, 8ax-mp 8 . 2  |-  ( ( ( ( ph  ->  ph )  ->  ( (  F.  ->  ph )  ->  ph )
)  ->  ( ( ph  ->  ph )  ->  ( ph  ->  ( ph  ->  ph ) ) ) )  ->  ( th  ->  ( ( th  ->  ph )  ->  ( ta  ->  ( ch  ->  ph ) ) ) ) )
101, 9ax-mp 8 1  |-  ( th 
->  ( ( th  ->  ph )  ->  ( ta  ->  ( ch  ->  ph )
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    F. wfal 1308
This theorem is referenced by:  mercolem6  1497  mercolem7  1498
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
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