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Theorem re1ax2lem 24821
Description: Lemma for re1ax2 24822. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
re1ax2lem  |-  ( (
ph  ->  ( ps  ->  ch ) )  ->  ( ps  ->  ( ph  ->  ch ) ) )

Proof of Theorem re1ax2lem
StepHypRef Expression
1 tb-ax2 24818 . . . 4  |-  ( ps 
->  ( ( ps  ->  ch )  ->  ps )
)
2 tb-ax1 24817 . . . 4  |-  ( ( ( ps  ->  ch )  ->  ps )  -> 
( ( ps  ->  ch )  ->  ( ( ps  ->  ch )  ->  ch ) ) )
31, 2tbsyl 24820 . . 3  |-  ( ps 
->  ( ( ps  ->  ch )  ->  ( ( ps  ->  ch )  ->  ch ) ) )
4 tb-ax1 24817 . . . 4  |-  ( ( ( ps  ->  ch )  ->  ( ( ps 
->  ch )  ->  ch ) )  ->  (
( ( ( ps 
->  ch )  ->  ch )  ->  ch )  -> 
( ( ps  ->  ch )  ->  ch )
) )
5 tb-ax3 24819 . . . 4  |-  ( ( ( ( ( ps 
->  ch )  ->  ch )  ->  ch )  -> 
( ( ps  ->  ch )  ->  ch )
)  ->  ( ( ps  ->  ch )  ->  ch ) )
64, 5tbsyl 24820 . . 3  |-  ( ( ( ps  ->  ch )  ->  ( ( ps 
->  ch )  ->  ch ) )  ->  (
( ps  ->  ch )  ->  ch ) )
73, 6tbsyl 24820 . 2  |-  ( ps 
->  ( ( ps  ->  ch )  ->  ch )
)
8 tb-ax1 24817 . 2  |-  ( (
ph  ->  ( ps  ->  ch ) )  ->  (
( ( ps  ->  ch )  ->  ch )  ->  ( ph  ->  ch ) ) )
9 tb-ax1 24817 . 2  |-  ( ( ps  ->  ( ( ps  ->  ch )  ->  ch ) )  ->  (
( ( ( ps 
->  ch )  ->  ch )  ->  ( ph  ->  ch ) )  ->  ( ps  ->  ( ph  ->  ch ) ) ) )
107, 8, 9mpsyl 59 1  |-  ( (
ph  ->  ( ps  ->  ch ) )  ->  ( ps  ->  ( ph  ->  ch ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4
This theorem is referenced by:  re1ax2  24822
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8
  Copyright terms: Public domain W3C validator