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Theorem necon2bbiddc 2316
Description: Contrapositive deduction for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
Hypothesis
Ref Expression
necon2bbiddc.1  |-  ( ph  ->  (DECID  A  =  B  -> 
( ps  <->  A  =/=  B ) ) )
Assertion
Ref Expression
necon2bbiddc  |-  ( ph  ->  (DECID  A  =  B  -> 
( A  =  B  <->  -.  ps ) ) )

Proof of Theorem necon2bbiddc
StepHypRef Expression
1 necon2bbiddc.1 . . . 4  |-  ( ph  ->  (DECID  A  =  B  -> 
( ps  <->  A  =/=  B ) ) )
2 bicom 138 . . . 4  |-  ( ( ps  <->  A  =/=  B
)  <->  ( A  =/= 
B  <->  ps ) )
31, 2syl6ib 159 . . 3  |-  ( ph  ->  (DECID  A  =  B  -> 
( A  =/=  B  <->  ps ) ) )
43necon1bbiddc 2312 . 2  |-  ( ph  ->  (DECID  A  =  B  -> 
( -.  ps  <->  A  =  B ) ) )
5 bicom 138 . 2  |-  ( ( -.  ps  <->  A  =  B )  <->  ( A  =  B  <->  -.  ps )
)
64, 5syl6ib 159 1  |-  ( ph  ->  (DECID  A  =  B  -> 
( A  =  B  <->  -.  ps ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 103  DECID wdc 776    = wceq 1285    =/= wne 2249
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663
This theorem depends on definitions:  df-bi 115  df-dc 777  df-ne 2250
This theorem is referenced by: (None)
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