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Metamath Proof Explorer

Theorem tbwlem1

Description: Used to rederive the Lukasiewicz axioms from Tarski-Bernays-Wajsberg'. (Contributed by Anthony Hart, 16-Aug-2011) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Assertion tbwlem1 
|- ( ( ph -> ( ps -> ch ) ) -> ( ps -> ( ph -> ch ) ) )

Proof

Step Hyp Ref Expression
1 tbw-ax1 
 |-  ( ( ph -> ( ps -> ch ) ) -> ( ( ( ps -> ch ) -> ch ) -> ( ph -> ch ) ) )
2 tbw-ax2 
 |-  ( ps -> ( ( ps -> ch ) -> ps ) )
3 tbw-ax1 
 |-  ( ( ( ps -> ch ) -> ps ) -> ( ( ps -> ch ) -> ( ( ps -> ch ) -> ch ) ) )
4 2 3 tbwsyl 
 |-  ( ps -> ( ( ps -> ch ) -> ( ( ps -> ch ) -> ch ) ) )
5 tbw-ax1 
 |-  ( ( ( ps -> ch ) -> ( ( ps -> ch ) -> ch ) ) -> ( ( ( ( ps -> ch ) -> ch ) -> ch ) -> ( ( ps -> ch ) -> ch ) ) )
6 tbw-ax3 
 |-  ( ( ( ( ( ps -> ch ) -> ch ) -> ch ) -> ( ( ps -> ch ) -> ch ) ) -> ( ( ps -> ch ) -> ch ) )
7 5 6 tbwsyl 
 |-  ( ( ( ps -> ch ) -> ( ( ps -> ch ) -> ch ) ) -> ( ( ps -> ch ) -> ch ) )
8 4 7 tbwsyl 
 |-  ( ps -> ( ( ps -> ch ) -> ch ) )
9 tbw-ax1 
 |-  ( ( ps -> ( ( ps -> ch ) -> ch ) ) -> ( ( ( ( ps -> ch ) -> ch ) -> ( ph -> ch ) ) -> ( ps -> ( ph -> ch ) ) ) )
10 8 9 ax-mp 
 |-  ( ( ( ( ps -> ch ) -> ch ) -> ( ph -> ch ) ) -> ( ps -> ( ph -> ch ) ) )
11 1 10 tbwsyl 
 |-  ( ( ph -> ( ps -> ch ) ) -> ( ps -> ( ph -> ch ) ) )