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Theorem con3ALTVD 28882
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 128). 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 O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. con3ALT 28469 is con3ALTVD 28882 without virtual deductions and was automatically derived from con3ALTVD 28882. 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 28519 . . . . . 6  |-  (. ( ph  ->  ps )  ->.  ( ph  ->  ps ) ).
2 idn2 28568 . . . . . . 7  |-  (. ( ph  ->  ps ) ,. 
-.  -.  ph  ->.  -.  -.  ph ).
3 notnot2 106 . . . . . . 7  |-  ( -. 
-.  ph  ->  ph )
42, 3e2 28586 . . . . . 6  |-  (. ( ph  ->  ps ) ,. 
-.  -.  ph  ->.  ph ).
5 id 20 . . . . . 6  |-  ( (
ph  ->  ps )  -> 
( ph  ->  ps )
)
61, 4, 5e12 28690 . . . . 5  |-  (. ( ph  ->  ps ) ,. 
-.  -.  ph  ->.  ps ).
7 notnot1 116 . . . . 5  |-  ( ps 
->  -.  -.  ps )
86, 7e2 28586 . . . 4  |-  (. ( ph  ->  ps ) ,. 
-.  -.  ph  ->.  -.  -.  ps ).
98in2 28560 . . 3  |-  (. ( ph  ->  ps )  ->.  ( -.  -.  ph  ->  -.  -.  ps ) ).
10 ax-3 7 . . 3  |-  ( ( -.  -.  ph  ->  -. 
-.  ps )  ->  ( -.  ps  ->  -.  ph )
)
119, 10e1_ 28582 . 2  |-  (. ( ph  ->  ps )  ->.  ( -.  ps  ->  -.  ph ) ).
1211in1 28516 1  |-  ( (
ph  ->  ps )  -> 
( -.  ps  ->  -. 
ph ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8
This theorem depends on definitions:  df-bi 178  df-an 361  df-vd1 28515  df-vd2 28524
  Copyright terms: Public domain W3C validator