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Theorem re1tbw1 1500
Description: tbw-ax1 1455 rederived from merco2 1491. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
re1tbw1  |-  ( (
ph  ->  ps )  -> 
( ( ps  ->  ch )  ->  ( ph  ->  ch ) ) )

Proof of Theorem re1tbw1
StepHypRef Expression
1 mercolem8 1499 . . 3  |-  ( (
ph  ->  ps )  -> 
( ( ps  ->  (
ph  ->  ch ) )  ->  ( ( ph  ->  ps )  ->  (
( ps  ->  ch )  ->  ( ph  ->  ch ) ) ) ) )
2 mercolem3 1494 . . 3  |-  ( ( ps  ->  ch )  ->  ( ps  ->  ( ph  ->  ch ) ) )
3 mercolem6 1497 . . 3  |-  ( ( ( ph  ->  ps )  ->  ( ( ps 
->  ( ph  ->  ch ) )  ->  (
( ph  ->  ps )  ->  ( ( ps  ->  ch )  ->  ( ph  ->  ch ) ) ) ) )  ->  (
( ps  ->  ( ph  ->  ch ) )  ->  ( ( ph  ->  ps )  ->  (
( ps  ->  ch )  ->  ( ph  ->  ch ) ) ) ) )
41, 2, 3mpsyl 59 . 2  |-  ( ( ps  ->  ch )  ->  ( ( ph  ->  ps )  ->  ( ( ps  ->  ch )  -> 
( ph  ->  ch )
) ) )
5 mercolem6 1497 . 2  |-  ( ( ( ps  ->  ch )  ->  ( ( ph  ->  ps )  ->  (
( ps  ->  ch )  ->  ( ph  ->  ch ) ) ) )  ->  ( ( ph  ->  ps )  ->  (
( ps  ->  ch )  ->  ( ph  ->  ch ) ) ) )
64, 5ax-mp 8 1  |-  ( (
ph  ->  ps )  -> 
( ( ps  ->  ch )  ->  ( ph  ->  ch ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4
This theorem is referenced by:  re1tbw4  1503
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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