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Theorem con5VD 33433
Description: Virtual deduction proof of con5 33025. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. con5 33025 is con5VD 33433 without virtual deductions and was automatically derived from con5VD 33433.
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 33084 . . . . 5  |-  (. ( ph 
<->  -.  ps )  ->.  ( ph  <->  -. 
ps ) ).
2 bi2 198 . . . . 5  |-  ( (
ph 
<->  -.  ps )  -> 
( -.  ps  ->  ph ) )
31, 2e1a 33146 . . . 4  |-  (. ( ph 
<->  -.  ps )  ->.  ( -.  ps  ->  ph ) ).
4 con3 134 . . . 4  |-  ( ( -.  ps  ->  ph )  ->  ( -.  ph  ->  -. 
-.  ps ) )
53, 4e1a 33146 . . 3  |-  (. ( ph 
<->  -.  ps )  ->.  ( -.  ph 
->  -.  -.  ps ) ).
6 notnot 291 . . 3  |-  ( ps  <->  -. 
-.  ps )
7 imbi2 324 . . . 4  |-  ( ( ps  <->  -.  -.  ps )  ->  ( ( -.  ph  ->  ps )  <->  ( -.  ph 
->  -.  -.  ps )
) )
87biimprcd 225 . . 3  |-  ( ( -.  ph  ->  -.  -.  ps )  ->  ( ( ps  <->  -.  -.  ps )  ->  ( -.  ph  ->  ps ) ) )
95, 6, 8e10 33213 . 2  |-  (. ( ph 
<->  -.  ps )  ->.  ( -.  ph 
->  ps ) ).
109in1 33081 1  |-  ( (
ph 
<->  -.  ps )  -> 
( -.  ph  ->  ps ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8
This theorem depends on definitions:  df-bi 185  df-vd1 33080
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator