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Metamath Proof Explorer

Theorem merco1lem1

Description: Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 . (Contributed by Anthony Hart, 17-Sep-2011) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Assertion merco1lem1 
|- ( ph -> ( F. -> ch ) )

Proof

Step Hyp Ref Expression
1 merco1 
 |-  ( ( ( ( ( F. -> ph ) -> ( ph -> F. ) ) -> ( ph -> F. ) ) -> ( F. -> ph ) ) -> ( ( ( F. -> ph ) -> F. ) -> ( ph -> F. ) ) )
2 merco1 
 |-  ( ( ( ( ( ( F. -> ph ) -> ( ph -> F. ) ) -> ( ph -> F. ) ) -> ( F. -> ph ) ) -> ( ( ( F. -> ph ) -> F. ) -> ( ph -> F. ) ) ) -> ( ( ( ( ( F. -> ph ) -> F. ) -> ( ph -> F. ) ) -> ( F. -> ph ) ) -> ( ph -> ( F. -> ph ) ) ) )
3 1 2 ax-mp 
 |-  ( ( ( ( ( F. -> ph ) -> F. ) -> ( ph -> F. ) ) -> ( F. -> ph ) ) -> ( ph -> ( F. -> ph ) ) )
4 merco1 
 |-  ( ( ( ( ( ( F. -> ph ) -> F. ) -> ( ph -> F. ) ) -> ( F. -> ph ) ) -> ( ph -> ( F. -> ph ) ) ) -> ( ( ( ph -> ( F. -> ph ) ) -> ( F. -> ph ) ) -> ( ph -> ( F. -> ph ) ) ) )
5 3 4 ax-mp 
 |-  ( ( ( ph -> ( F. -> ph ) ) -> ( F. -> ph ) ) -> ( ph -> ( F. -> ph ) ) )
6 merco1 
 |-  ( ( ( ( ( F. -> ph ) -> ( ph -> F. ) ) -> ( ( ph -> ( F. -> ph ) ) -> F. ) ) -> ( ph -> ( F. -> ph ) ) ) -> ( ( ( ph -> ( F. -> ph ) ) -> F. ) -> ( ph -> F. ) ) )
7 merco1 
 |-  ( ( ( ( ( ( F. -> ph ) -> ( ph -> F. ) ) -> ( ( ph -> ( F. -> ph ) ) -> F. ) ) -> ( ph -> ( F. -> ph ) ) ) -> ( ( ( ph -> ( F. -> ph ) ) -> F. ) -> ( ph -> F. ) ) ) -> ( ( ( ( ( ph -> ( F. -> ph ) ) -> F. ) -> ( ph -> F. ) ) -> ( F. -> ph ) ) -> ( ( ph -> ( F. -> ph ) ) -> ( F. -> ph ) ) ) )
8 6 7 ax-mp 
 |-  ( ( ( ( ( ph -> ( F. -> ph ) ) -> F. ) -> ( ph -> F. ) ) -> ( F. -> ph ) ) -> ( ( ph -> ( F. -> ph ) ) -> ( F. -> ph ) ) )
9 merco1 
 |-  ( ( ( ( ( ( ph -> ( F. -> ph ) ) -> F. ) -> ( ph -> F. ) ) -> ( F. -> ph ) ) -> ( ( ph -> ( F. -> ph ) ) -> ( F. -> ph ) ) ) -> ( ( ( ( ph -> ( F. -> ph ) ) -> ( F. -> ph ) ) -> ( ph -> ( F. -> ph ) ) ) -> ( ph -> ( ph -> ( F. -> ph ) ) ) ) )
10 8 9 ax-mp 
 |-  ( ( ( ( ph -> ( F. -> ph ) ) -> ( F. -> ph ) ) -> ( ph -> ( F. -> ph ) ) ) -> ( ph -> ( ph -> ( F. -> ph ) ) ) )
11 5 10 ax-mp 
 |-  ( ph -> ( ph -> ( F. -> ph ) ) )
12 merco1 
 |-  ( ( ( ( ( F. -> ph ) -> ( ph -> F. ) ) -> ( ph -> F. ) ) -> ( F. -> ch ) ) -> ( ( ( F. -> ch ) -> F. ) -> ( ph -> F. ) ) )
13 merco1 
 |-  ( ( ( ( ( ( F. -> ph ) -> ( ph -> F. ) ) -> ( ph -> F. ) ) -> ( F. -> ch ) ) -> ( ( ( F. -> ch ) -> F. ) -> ( ph -> F. ) ) ) -> ( ( ( ( ( F. -> ch ) -> F. ) -> ( ph -> F. ) ) -> ( F. -> ph ) ) -> ( ph -> ( F. -> ph ) ) ) )
14 12 13 ax-mp 
 |-  ( ( ( ( ( F. -> ch ) -> F. ) -> ( ph -> F. ) ) -> ( F. -> ph ) ) -> ( ph -> ( F. -> ph ) ) )
15 merco1 
 |-  ( ( ( ( ( ( F. -> ch ) -> F. ) -> ( ph -> F. ) ) -> ( F. -> ph ) ) -> ( ph -> ( F. -> ph ) ) ) -> ( ( ( ph -> ( F. -> ph ) ) -> ( F. -> ch ) ) -> ( ph -> ( F. -> ch ) ) ) )
16 14 15 ax-mp 
 |-  ( ( ( ph -> ( F. -> ph ) ) -> ( F. -> ch ) ) -> ( ph -> ( F. -> ch ) ) )
17 merco1 
 |-  ( ( ( ( ( F. -> ch ) -> ( ( ph -> ( ph -> ( F. -> ph ) ) ) -> F. ) ) -> ( ( ph -> ( F. -> ph ) ) -> F. ) ) -> ( ph -> ( F. -> ch ) ) ) -> ( ( ( ph -> ( F. -> ch ) ) -> F. ) -> ( ( ph -> ( ph -> ( F. -> ph ) ) ) -> F. ) ) )
18 merco1 
 |-  ( ( ( ( ( ( F. -> ch ) -> ( ( ph -> ( ph -> ( F. -> ph ) ) ) -> F. ) ) -> ( ( ph -> ( F. -> ph ) ) -> F. ) ) -> ( ph -> ( F. -> ch ) ) ) -> ( ( ( ph -> ( F. -> ch ) ) -> F. ) -> ( ( ph -> ( ph -> ( F. -> ph ) ) ) -> F. ) ) ) -> ( ( ( ( ( ph -> ( F. -> ch ) ) -> F. ) -> ( ( ph -> ( ph -> ( F. -> ph ) ) ) -> F. ) ) -> ( F. -> ch ) ) -> ( ( ph -> ( F. -> ph ) ) -> ( F. -> ch ) ) ) )
19 17 18 ax-mp 
 |-  ( ( ( ( ( ph -> ( F. -> ch ) ) -> F. ) -> ( ( ph -> ( ph -> ( F. -> ph ) ) ) -> F. ) ) -> ( F. -> ch ) ) -> ( ( ph -> ( F. -> ph ) ) -> ( F. -> ch ) ) )
20 merco1 
 |-  ( ( ( ( ( ( ph -> ( F. -> ch ) ) -> F. ) -> ( ( ph -> ( ph -> ( F. -> ph ) ) ) -> F. ) ) -> ( F. -> ch ) ) -> ( ( ph -> ( F. -> ph ) ) -> ( F. -> ch ) ) ) -> ( ( ( ( ph -> ( F. -> ph ) ) -> ( F. -> ch ) ) -> ( ph -> ( F. -> ch ) ) ) -> ( ( ph -> ( ph -> ( F. -> ph ) ) ) -> ( ph -> ( F. -> ch ) ) ) ) )
21 19 20 ax-mp 
 |-  ( ( ( ( ph -> ( F. -> ph ) ) -> ( F. -> ch ) ) -> ( ph -> ( F. -> ch ) ) ) -> ( ( ph -> ( ph -> ( F. -> ph ) ) ) -> ( ph -> ( F. -> ch ) ) ) )
22 16 21 ax-mp 
 |-  ( ( ph -> ( ph -> ( F. -> ph ) ) ) -> ( ph -> ( F. -> ch ) ) )
23 11 22 ax-mp 
 |-  ( ph -> ( F. -> ch ) )