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Metamath Proof Explorer

Theorem retbwax2

Description: tbw-ax2 rederived from merco1 . (Contributed by Anthony Hart, 17-Sep-2011) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Assertion retbwax2 
|- ( ph -> ( ps -> ph ) )

Proof

Step Hyp Ref Expression
1 merco1lem1 
 |-  ( ( ( ( ( ph -> ph ) -> ph ) -> ( ph -> F. ) ) -> ph ) -> ( F. -> ph ) )
2 merco1 
 |-  ( ( ( ( ( ( ph -> ph ) -> ph ) -> ( ph -> F. ) ) -> ph ) -> ( F. -> ph ) ) -> ( ( ( F. -> ph ) -> ( ph -> ph ) ) -> ( ph -> ( ph -> ph ) ) ) )
3 1 2 ax-mp 
 |-  ( ( ( F. -> ph ) -> ( ph -> ph ) ) -> ( ph -> ( ph -> ph ) ) )
4 merco1 
 |-  ( ( ( ( ( ph -> ( ph -> ph ) ) -> ( ph -> F. ) ) -> ( ph -> F. ) ) -> F. ) -> ( ( F. -> ph ) -> ( ph -> ph ) ) )
5 merco1 
 |-  ( ( ( ( ( ( ph -> ( ph -> ph ) ) -> ( ph -> F. ) ) -> ( ph -> F. ) ) -> F. ) -> ( ( F. -> ph ) -> ( ph -> ph ) ) ) -> ( ( ( ( F. -> ph ) -> ( ph -> ph ) ) -> ( ph -> ( ph -> ph ) ) ) -> ( ph -> ( ph -> ( ph -> ph ) ) ) ) )
6 4 5 ax-mp 
 |-  ( ( ( ( F. -> ph ) -> ( ph -> ph ) ) -> ( ph -> ( ph -> ph ) ) ) -> ( ph -> ( ph -> ( ph -> ph ) ) ) )
7 3 6 ax-mp 
 |-  ( ph -> ( ph -> ( ph -> ph ) ) )
8 merco1lem1 
 |-  ( ( ( ( ( ps -> ph ) -> ph ) -> ( ph -> F. ) ) -> ph ) -> ( F. -> ph ) )
9 merco1 
 |-  ( ( ( ( ( ( ps -> ph ) -> ph ) -> ( ph -> F. ) ) -> ph ) -> ( F. -> ph ) ) -> ( ( ( F. -> ph ) -> ( ps -> ph ) ) -> ( ph -> ( ps -> ph ) ) ) )
10 8 9 ax-mp 
 |-  ( ( ( F. -> ph ) -> ( ps -> ph ) ) -> ( ph -> ( ps -> ph ) ) )
11 merco1 
 |-  ( ( ( ( ( ph -> ( ps -> ph ) ) -> ( ps -> F. ) ) -> ( ( ph -> ( ph -> ( ph -> ph ) ) ) -> F. ) ) -> F. ) -> ( ( F. -> ph ) -> ( ps -> ph ) ) )
12 merco1 
 |-  ( ( ( ( ( ( ph -> ( ps -> ph ) ) -> ( ps -> F. ) ) -> ( ( ph -> ( ph -> ( ph -> ph ) ) ) -> F. ) ) -> F. ) -> ( ( F. -> ph ) -> ( ps -> ph ) ) ) -> ( ( ( ( F. -> ph ) -> ( ps -> ph ) ) -> ( ph -> ( ps -> ph ) ) ) -> ( ( ph -> ( ph -> ( ph -> ph ) ) ) -> ( ph -> ( ps -> ph ) ) ) ) )
13 11 12 ax-mp 
 |-  ( ( ( ( F. -> ph ) -> ( ps -> ph ) ) -> ( ph -> ( ps -> ph ) ) ) -> ( ( ph -> ( ph -> ( ph -> ph ) ) ) -> ( ph -> ( ps -> ph ) ) ) )
14 10 13 ax-mp 
 |-  ( ( ph -> ( ph -> ( ph -> ph ) ) ) -> ( ph -> ( ps -> ph ) ) )
15 7 14 ax-mp 
 |-  ( ph -> ( ps -> ph ) )