Metamath Proof Explorer

Theorem re1ax2

Description: ax-2 rederived from the Tarski-Bernays axiom system. Often tb-ax1 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.)

Ref Expression
Assertion re1ax2 
|- ( ( ph -> ( ps -> ch ) ) -> ( ( ph -> ps ) -> ( ph -> ch ) ) )

Proof

Step Hyp Ref Expression
1 re1ax2lem 
 |-  ( ( ph -> ( ps -> ch ) ) -> ( ps -> ( ph -> ch ) ) )
2 tb-ax1 
 |-  ( ( ph -> ( ph -> ch ) ) -> ( ( ( ph -> ch ) -> ch ) -> ( ph -> ch ) ) )
3 tb-ax3 
 |-  ( ( ( ( ph -> ch ) -> ch ) -> ( ph -> ch ) ) -> ( ph -> ch ) )
4 2 3 tbsyl 
 |-  ( ( ph -> ( ph -> ch ) ) -> ( ph -> ch ) )
5 tb-ax1 
 |-  ( ( ph -> ps ) -> ( ( ps -> ( ph -> ch ) ) -> ( ph -> ( ph -> ch ) ) ) )
6 re1ax2lem 
 |-  ( ( ( ph -> ps ) -> ( ( ps -> ( ph -> ch ) ) -> ( ph -> ( ph -> ch ) ) ) ) -> ( ( ps -> ( ph -> ch ) ) -> ( ( ph -> ps ) -> ( ph -> ( ph -> ch ) ) ) ) )
7 5 6 ax-mp 
 |-  ( ( ps -> ( ph -> ch ) ) -> ( ( ph -> ps ) -> ( ph -> ( ph -> ch ) ) ) )
8 tb-ax1 
 |-  ( ( ( ph -> ps ) -> ( ph -> ( ph -> ch ) ) ) -> ( ( ( ph -> ( ph -> ch ) ) -> ( ph -> ch ) ) -> ( ( ph -> ps ) -> ( ph -> ch ) ) ) )
9 re1ax2lem 
 |-  ( ( ( ( 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 ) ) ) ) )
10 8 9 ax-mp 
 |-  ( ( ( ph -> ( ph -> ch ) ) -> ( ph -> ch ) ) -> ( ( ( ph -> ps ) -> ( ph -> ( ph -> ch ) ) ) -> ( ( ph -> ps ) -> ( ph -> ch ) ) ) )
11 4 7 10 mpsyl 
 |-  ( ( ps -> ( ph -> ch ) ) -> ( ( ph -> ps ) -> ( ph -> ch ) ) )
12 1 11 tbsyl 
 |-  ( ( ph -> ( ps -> ch ) ) -> ( ( ph -> ps ) -> ( ph -> ch ) ) )