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Theorem con3ALTVD 33197
Description: The following User's Proof is a Natural Deduction Sequent Calculus transcription of the Fitch-style Natural Deduction proof of Theorem 7 of Section 14 of [Margaris] p. 60 ( which is con3 134). The same proof may also be interpreted to be a Virtual Deduction Hilbert-style axiomatic proof. It was completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. con3ALT 32780 is con3ALTVD 33197 without virtual deductions and was automatically derived from con3ALTVD 33197. Step i of the User's Proof corresponds to step i of the Fitch-style proof.
1::  |-  (. ( ph  ->  ps )  ->.  ( ph  ->  ps ) ).
2::  |-  (. ( ph  ->  ps ) ,. -.  -.  ph  ->.  -.  -.  ph ).
3::  |-  ( -.  -.  ph  ->  ph )
4:2:  |-  (. ( ph  ->  ps ) ,. -.  -.  ph  ->.  ph ).
5:1,4:  |-  (. ( ph  ->  ps ) ,. -.  -.  ph  ->.  ps ).
6::  |-  ( ps  ->  -.  -.  ps )
7:6,5:  |-  (. ( ph  ->  ps ) ,. -.  -.  ph  ->.  -.  -.  ps ).
8:7:  |-  (. ( ph  ->  ps )  ->.  ( -.  -.  ph  ->  -.  -.  ps  ) ).
9::  |-  ( ( -.  -.  ph  ->  -.  -.  ps )  ->  ( -.  ps  ->  -.  ph ) )
10:8:  |-  (. ( ph  ->  ps )  ->.  ( -.  ps  ->  -.  ph ) ).
qed:10:  |-  ( ( ph  ->  ps )  ->  ( -.  ps  ->  -.  ph ) )
(Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
con3ALTVD  |-  ( (
ph  ->  ps )  -> 
( -.  ps  ->  -. 
ph ) )

Proof of Theorem con3ALTVD
StepHypRef Expression
1 idn1 32832 . . . . . 6  |-  (. ( ph  ->  ps )  ->.  ( ph  ->  ps ) ).
2 idn2 32880 . . . . . . 7  |-  (. ( ph  ->  ps ) ,. 
-.  -.  ph  ->.  -.  -.  ph ).
3 notnot2 112 . . . . . . 7  |-  ( -. 
-.  ph  ->  ph )
42, 3e2 32898 . . . . . 6  |-  (. ( ph  ->  ps ) ,. 
-.  -.  ph  ->.  ph ).
5 id 22 . . . . . 6  |-  ( (
ph  ->  ps )  -> 
( ph  ->  ps )
)
61, 4, 5e12 33002 . . . . 5  |-  (. ( ph  ->  ps ) ,. 
-.  -.  ph  ->.  ps ).
7 notnot1 122 . . . . 5  |-  ( ps 
->  -.  -.  ps )
86, 7e2 32898 . . . 4  |-  (. ( ph  ->  ps ) ,. 
-.  -.  ph  ->.  -.  -.  ps ).
98in2 32872 . . 3  |-  (. ( ph  ->  ps )  ->.  ( -.  -.  ph  ->  -.  -.  ps ) ).
10 ax-3 8 . . 3  |-  ( ( -.  -.  ph  ->  -. 
-.  ps )  ->  ( -.  ps  ->  -.  ph )
)
119, 10e1a 32894 . 2  |-  (. ( ph  ->  ps )  ->.  ( -.  ps  ->  -.  ph ) ).
1211in1 32829 1  |-  ( (
ph  ->  ps )  -> 
( -.  ps  ->  -. 
ph ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8
This theorem depends on definitions:  df-bi 185  df-an 371  df-vd1 32828  df-vd2 32836
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator