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Theorem con3ALTVD 28692
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 126). 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 28293 is con3ALTVD 28692 without virtual deductions and was automatically derived from con3ALTVD 28692. 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 28342 . . . . . 6  |-  (. ( ph  ->  ps )  ->.  ( ph  ->  ps ) ).
2 idn2 28385 . . . . . . 7  |-  (. ( ph  ->  ps ) ,. 
-.  -.  ph  ->.  -.  -.  ph ).
3 notnot2 104 . . . . . . 7  |-  ( -. 
-.  ph  ->  ph )
42, 3e2 28403 . . . . . 6  |-  (. ( ph  ->  ps ) ,. 
-.  -.  ph  ->.  ph ).
5 id 19 . . . . . 6  |-  ( (
ph  ->  ps )  -> 
( ph  ->  ps )
)
61, 4, 5e12 28499 . . . . 5  |-  (. ( ph  ->  ps ) ,. 
-.  -.  ph  ->.  ps ).
7 notnot1 114 . . . . 5  |-  ( ps 
->  -.  -.  ps )
86, 7e2 28403 . . . 4  |-  (. ( ph  ->  ps ) ,. 
-.  -.  ph  ->.  -.  -.  ps ).
98in2 28377 . . 3  |-  (. ( ph  ->  ps )  ->.  ( -.  -.  ph  ->  -.  -.  ps ) ).
10 ax-3 7 . . 3  |-  ( ( -.  -.  ph  ->  -. 
-.  ps )  ->  ( -.  ps  ->  -.  ph )
)
119, 10e1_ 28399 . 2  |-  (. ( ph  ->  ps )  ->.  ( -.  ps  ->  -.  ph ) ).
1211in1 28339 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 177  df-an 360  df-vd1 28338  df-vd2 28347
  Copyright terms: Public domain W3C validator