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Theorem necon1abiddc 2282
Description: Contrapositive deduction for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
Hypothesis
Ref Expression
necon1abiddc.1  |-  ( ph  ->  (DECID  ps  ->  ( -.  ps 
<->  A  =  B ) ) )
Assertion
Ref Expression
necon1abiddc  |-  ( ph  ->  (DECID  ps  ->  ( A  =/=  B  <->  ps ) ) )

Proof of Theorem necon1abiddc
StepHypRef Expression
1 necon1abiddc.1 . . 3  |-  ( ph  ->  (DECID  ps  ->  ( -.  ps 
<->  A  =  B ) ) )
21con1biddc 781 . 2  |-  ( ph  ->  (DECID  ps  ->  ( -.  A  =  B  <->  ps )
) )
3 df-ne 2221 . . 3  |-  ( A  =/=  B  <->  -.  A  =  B )
43bibi1i 221 . 2  |-  ( ( A  =/=  B  <->  ps )  <->  ( -.  A  =  B  <->  ps ) )
52, 4syl6ibr 155 1  |-  ( ph  ->  (DECID  ps  ->  ( A  =/=  B  <->  ps ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 102  DECID wdc 753    = wceq 1259    =/= wne 2220
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640
This theorem depends on definitions:  df-bi 114  df-dc 754  df-ne 2221
This theorem is referenced by:  necon2abiddc  2286
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