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Metamath Proof Explorer

Theorem tbwlem5

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 tbwlem5 
|- ( ( ( ph -> ( ps -> F. ) ) -> F. ) -> ph )

Proof

Step Hyp Ref Expression
1 tbw-ax2 
 |-  ( ph -> ( ps -> ph ) )
2 tbw-ax1 
 |-  ( ( ps -> ph ) -> ( ( ph -> F. ) -> ( ps -> F. ) ) )
3 1 2 tbwsyl 
 |-  ( ph -> ( ( ph -> F. ) -> ( ps -> F. ) ) )
4 tbwlem1 
 |-  ( ( ph -> ( ( ph -> F. ) -> ( ps -> F. ) ) ) -> ( ( ph -> F. ) -> ( ph -> ( ps -> F. ) ) ) )
5 3 4 ax-mp 
 |-  ( ( ph -> F. ) -> ( ph -> ( ps -> F. ) ) )
6 tbwlem4 
 |-  ( ( ( ph -> F. ) -> ( ph -> ( ps -> F. ) ) ) -> ( ( ( ph -> ( ps -> F. ) ) -> F. ) -> ph ) )
7 5 6 ax-mp 
 |-  ( ( ( ph -> ( ps -> F. ) ) -> F. ) -> ph )