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Theorem con5VD 29012
Description: Virtual deduction proof of con5 28606. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. con5 28606 is con5VD 29012 without virtual deductions and was automatically derived from con5VD 29012.
1::  |-  (. ( ph  <->  -.  ps )  ->.  ( ph  <->  -.  ps ) ).
2:1:  |-  (. ( ph  <->  -.  ps )  ->.  ( -.  ps  ->  ph ) ).
3:2:  |-  (. ( ph  <->  -.  ps )  ->.  ( -.  ph  ->  -.  -.  ps  ) ).
4::  |-  ( ps  <->  -.  -.  ps )
5:3,4:  |-  (. ( ph  <->  -.  ps )  ->.  ( -.  ph  ->  ps ) ).
qed:5:  |-  ( ( ph  <->  -.  ps )  ->  ( -.  ph  ->  ps ) )
(Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
con5VD  |-  ( (
ph 
<->  -.  ps )  -> 
( -.  ph  ->  ps ) )

Proof of Theorem con5VD
StepHypRef Expression
1 idn1 28665 . . . . 5  |-  (. ( ph 
<->  -.  ps )  ->.  ( ph  <->  -. 
ps ) ).
2 bi2 190 . . . . 5  |-  ( (
ph 
<->  -.  ps )  -> 
( -.  ps  ->  ph ) )
31, 2e1_ 28728 . . . 4  |-  (. ( ph 
<->  -.  ps )  ->.  ( -.  ps  ->  ph ) ).
4 con3 128 . . . 4  |-  ( ( -.  ps  ->  ph )  ->  ( -.  ph  ->  -. 
-.  ps ) )
53, 4e1_ 28728 . . 3  |-  (. ( ph 
<->  -.  ps )  ->.  ( -.  ph 
->  -.  -.  ps ) ).
6 notnot 283 . . 3  |-  ( ps  <->  -. 
-.  ps )
7 imbi2 315 . . . 4  |-  ( ( ps  <->  -.  -.  ps )  ->  ( ( -.  ph  ->  ps )  <->  ( -.  ph 
->  -.  -.  ps )
) )
87biimprcd 217 . . 3  |-  ( ( -.  ph  ->  -.  -.  ps )  ->  ( ( ps  <->  -.  -.  ps )  ->  ( -.  ph  ->  ps ) ) )
95, 6, 8e10 28795 . 2  |-  (. ( ph 
<->  -.  ps )  ->.  ( -.  ph 
->  ps ) ).
109in1 28662 1  |-  ( (
ph 
<->  -.  ps )  -> 
( -.  ph  ->  ps ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177
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-vd1 28661
  Copyright terms: Public domain W3C validator