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Metamath Proof Explorer

Theorem re1ax2lem

Description: Lemma for re1ax2 . (Contributed by Anthony Hart, 16-Aug-2011) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Assertion re1ax2lem 
|- ( ( ph -> ( ps -> ch ) ) -> ( ps -> ( ph -> ch ) ) )

Proof

Step Hyp Ref Expression
1 tb-ax2 
 |-  ( ps -> ( ( ps -> ch ) -> ps ) )
2 tb-ax1 
 |-  ( ( ( ps -> ch ) -> ps ) -> ( ( ps -> ch ) -> ( ( ps -> ch ) -> ch ) ) )
3 1 2 tbsyl 
 |-  ( ps -> ( ( ps -> ch ) -> ( ( ps -> ch ) -> ch ) ) )
4 tb-ax1 
 |-  ( ( ( ps -> ch ) -> ( ( ps -> ch ) -> ch ) ) -> ( ( ( ( ps -> ch ) -> ch ) -> ch ) -> ( ( ps -> ch ) -> ch ) ) )
5 tb-ax3 
 |-  ( ( ( ( ( ps -> ch ) -> ch ) -> ch ) -> ( ( ps -> ch ) -> ch ) ) -> ( ( ps -> ch ) -> ch ) )
6 4 5 tbsyl 
 |-  ( ( ( ps -> ch ) -> ( ( ps -> ch ) -> ch ) ) -> ( ( ps -> ch ) -> ch ) )
7 3 6 tbsyl 
 |-  ( ps -> ( ( ps -> ch ) -> ch ) )
8 tb-ax1 
 |-  ( ( ph -> ( ps -> ch ) ) -> ( ( ( ps -> ch ) -> ch ) -> ( ph -> ch ) ) )
9 tb-ax1 
 |-  ( ( ps -> ( ( ps -> ch ) -> ch ) ) -> ( ( ( ( ps -> ch ) -> ch ) -> ( ph -> ch ) ) -> ( ps -> ( ph -> ch ) ) ) )
10 7 8 9 mpsyl 
 |-  ( ( ph -> ( ps -> ch ) ) -> ( ps -> ( ph -> ch ) ) )