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Theorem con3ALTVD 27961
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 27565 is con3ALTVD 27961 without virtual deductions and was automatically derived from con3ALTVD 27961. 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 27614 . . . . . 6  |-  (. ( ph  ->  ps )  ->.  ( ph  ->  ps ) ).
2 idn2 27654 . . . . . . 7  |-  (. ( ph  ->  ps ) ,. 
-.  -.  ph  ->.  -.  -.  ph ).
3 notnot2 106 . . . . . . 7  |-  ( -. 
-.  ph  ->  ph )
42, 3e2 27672 . . . . . 6  |-  (. ( ph  ->  ps ) ,. 
-.  -.  ph  ->.  ph ).
5 id 21 . . . . . 6  |-  ( (
ph  ->  ps )  -> 
( ph  ->  ps )
)
61, 4, 5e12 27768 . . . . 5  |-  (. ( ph  ->  ps ) ,. 
-.  -.  ph  ->.  ps ).
7 notnot1 116 . . . . 5  |-  ( ps 
->  -.  -.  ps )
86, 7e2 27672 . . . 4  |-  (. ( ph  ->  ps ) ,. 
-.  -.  ph  ->.  -.  -.  ps ).
98in2 27646 . . 3  |-  (. ( ph  ->  ps )  ->.  ( -.  -.  ph  ->  -.  -.  ps ) ).
10 ax-3 9 . . 3  |-  ( ( -.  -.  ph  ->  -. 
-.  ps )  ->  ( -.  ps  ->  -.  ph )
)
119, 10e1_ 27668 . 2  |-  (. ( ph  ->  ps )  ->.  ( -.  ps  ->  -.  ph ) ).
1211in1 27611 1  |-  ( (
ph  ->  ps )  -> 
( -.  ps  ->  -. 
ph ) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10
This theorem depends on definitions:  df-bi 179  df-an 362  df-vd1 27610  df-vd2 27619
  Copyright terms: Public domain W3C validator