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Theorem con3ALTVD 33584
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 33168 is con3ALTVD 33584 without virtual deductions and was automatically derived from con3ALTVD 33584. 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 33219 . . . . . 6  |-  (. ( ph  ->  ps )  ->.  ( ph  ->  ps ) ).
2 idn2 33267 . . . . . . 7  |-  (. ( ph  ->  ps ) ,. 
-.  -.  ph  ->.  -.  -.  ph ).
3 notnot2 112 . . . . . . 7  |-  ( -. 
-.  ph  ->  ph )
42, 3e2 33285 . . . . . 6  |-  (. ( ph  ->  ps ) ,. 
-.  -.  ph  ->.  ph ).
5 id 22 . . . . . 6  |-  ( (
ph  ->  ps )  -> 
( ph  ->  ps )
)
61, 4, 5e12 33389 . . . . 5  |-  (. ( ph  ->  ps ) ,. 
-.  -.  ph  ->.  ps ).
7 notnot1 122 . . . . 5  |-  ( ps 
->  -.  -.  ps )
86, 7e2 33285 . . . 4  |-  (. ( ph  ->  ps ) ,. 
-.  -.  ph  ->.  -.  -.  ps ).
98in2 33259 . . 3  |-  (. ( ph  ->  ps )  ->.  ( -.  -.  ph  ->  -.  -.  ps ) ).
10 ax-3 8 . . 3  |-  ( ( -.  -.  ph  ->  -. 
-.  ps )  ->  ( -.  ps  ->  -.  ph )
)
119, 10e1a 33281 . 2  |-  (. ( ph  ->  ps )  ->.  ( -.  ps  ->  -.  ph ) ).
1211in1 33216 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 33215  df-vd2 33223
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator