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Theorem re1ax2 24894
Description: ax-2 6 rederived from the Tarski-Bernays axiom system. Often tb-ax1 24889 is replaced with this theorem to make a "standard" system. This is because this theorem is easier to work with, despite it being longer. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
re1ax2  |-  ( (
ph  ->  ( ps  ->  ch ) )  ->  (
( ph  ->  ps )  ->  ( ph  ->  ch ) ) )

Proof of Theorem re1ax2
StepHypRef Expression
1 re1ax2lem 24893 . 2  |-  ( (
ph  ->  ( ps  ->  ch ) )  ->  ( ps  ->  ( ph  ->  ch ) ) )
2 tb-ax1 24889 . . . 4  |-  ( (
ph  ->  ( ph  ->  ch ) )  ->  (
( ( ph  ->  ch )  ->  ch )  ->  ( ph  ->  ch ) ) )
3 tb-ax3 24891 . . . 4  |-  ( ( ( ( ph  ->  ch )  ->  ch )  ->  ( ph  ->  ch ) )  ->  ( ph  ->  ch ) )
42, 3tbsyl 24892 . . 3  |-  ( (
ph  ->  ( ph  ->  ch ) )  ->  ( ph  ->  ch ) )
5 tb-ax1 24889 . . . 4  |-  ( (
ph  ->  ps )  -> 
( ( ps  ->  (
ph  ->  ch ) )  ->  ( ph  ->  (
ph  ->  ch ) ) ) )
6 re1ax2lem 24893 . . . 4  |-  ( ( ( ph  ->  ps )  ->  ( ( ps 
->  ( ph  ->  ch ) )  ->  ( ph  ->  ( ph  ->  ch ) ) ) )  ->  ( ( ps 
->  ( ph  ->  ch ) )  ->  (
( ph  ->  ps )  ->  ( ph  ->  ( ph  ->  ch ) ) ) ) )
75, 6ax-mp 8 . . 3  |-  ( ( ps  ->  ( ph  ->  ch ) )  -> 
( ( ph  ->  ps )  ->  ( ph  ->  ( ph  ->  ch ) ) ) )
8 tb-ax1 24889 . . . 4  |-  ( ( ( ph  ->  ps )  ->  ( ph  ->  (
ph  ->  ch ) ) )  ->  ( (
( ph  ->  ( ph  ->  ch ) )  -> 
( ph  ->  ch )
)  ->  ( ( ph  ->  ps )  -> 
( ph  ->  ch )
) ) )
9 re1ax2lem 24893 . . . 4  |-  ( ( ( ( ph  ->  ps )  ->  ( ph  ->  ( ph  ->  ch ) ) )  -> 
( ( ( ph  ->  ( ph  ->  ch ) )  ->  ( ph  ->  ch ) )  ->  ( ( ph  ->  ps )  ->  ( ph  ->  ch ) ) ) )  ->  (
( ( ph  ->  (
ph  ->  ch ) )  ->  ( ph  ->  ch ) )  ->  (
( ( ph  ->  ps )  ->  ( ph  ->  ( ph  ->  ch ) ) )  -> 
( ( ph  ->  ps )  ->  ( ph  ->  ch ) ) ) ) )
108, 9ax-mp 8 . . 3  |-  ( ( ( ph  ->  ( ph  ->  ch ) )  ->  ( ph  ->  ch ) )  ->  (
( ( ph  ->  ps )  ->  ( ph  ->  ( ph  ->  ch ) ) )  -> 
( ( ph  ->  ps )  ->  ( ph  ->  ch ) ) ) )
114, 7, 10mpsyl 59 . 2  |-  ( ( ps  ->  ( ph  ->  ch ) )  -> 
( ( ph  ->  ps )  ->  ( ph  ->  ch ) ) )
121, 11tbsyl 24892 1  |-  ( (
ph  ->  ( ps  ->  ch ) )  ->  (
( ph  ->  ps )  ->  ( ph  ->  ch ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8
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