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Theorem con5VD 28676
Description: Virtual deduction proof of con5 28285. 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 28285 is con5VD 28676 without virtual deductions and was automatically derived from con5VD 28676.
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 28342 . . . . 5  |-  (. ( ph 
<->  -.  ps )  ->.  ( ph  <->  -. 
ps ) ).
2 bi2 189 . . . . 5  |-  ( (
ph 
<->  -.  ps )  -> 
( -.  ps  ->  ph ) )
31, 2e1_ 28399 . . . 4  |-  (. ( ph 
<->  -.  ps )  ->.  ( -.  ps  ->  ph ) ).
4 con3 126 . . . 4  |-  ( ( -.  ps  ->  ph )  ->  ( -.  ph  ->  -. 
-.  ps ) )
53, 4e1_ 28399 . . 3  |-  (. ( ph 
<->  -.  ps )  ->.  ( -.  ph 
->  -.  -.  ps ) ).
6 notnot 282 . . 3  |-  ( ps  <->  -. 
-.  ps )
7 imbi2 314 . . . 4  |-  ( ( ps  <->  -.  -.  ps )  ->  ( ( -.  ph  ->  ps )  <->  ( -.  ph 
->  -.  -.  ps )
) )
87biimprcd 216 . . 3  |-  ( ( -.  ph  ->  -.  -.  ps )  ->  ( ( ps  <->  -.  -.  ps )  ->  ( -.  ph  ->  ps ) ) )
95, 6, 8e10 28467 . 2  |-  (. ( ph 
<->  -.  ps )  ->.  ( -.  ph 
->  ps ) ).
109in1 28339 1  |-  ( (
ph 
<->  -.  ps )  -> 
( -.  ph  ->  ps ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176
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-vd1 28338
  Copyright terms: Public domain W3C validator