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Theorem necon4biddc 2330
Description: Contrapositive law deduction for inequality. (Contributed by Jim Kingdon, 19-May-2018.)
Hypothesis
Ref Expression
necon4biddc.1  |-  ( ph  ->  (DECID  A  =  B  -> 
(DECID 
C  =  D  -> 
( A  =/=  B  <->  C  =/=  D ) ) ) )
Assertion
Ref Expression
necon4biddc  |-  ( ph  ->  (DECID  A  =  B  -> 
(DECID 
C  =  D  -> 
( A  =  B  <-> 
C  =  D ) ) ) )

Proof of Theorem necon4biddc
StepHypRef Expression
1 necon4biddc.1 . . 3  |-  ( ph  ->  (DECID  A  =  B  -> 
(DECID 
C  =  D  -> 
( A  =/=  B  <->  C  =/=  D ) ) ) )
2 df-ne 2256 . . . 4  |-  ( C  =/=  D  <->  -.  C  =  D )
32bibi2i 225 . . 3  |-  ( ( A  =/=  B  <->  C  =/=  D )  <->  ( A  =/= 
B  <->  -.  C  =  D ) )
41, 3syl8ib 164 . 2  |-  ( ph  ->  (DECID  A  =  B  -> 
(DECID 
C  =  D  -> 
( A  =/=  B  <->  -.  C  =  D ) ) ) )
54necon4abiddc 2328 1  |-  ( ph  ->  (DECID  A  =  B  -> 
(DECID 
C  =  D  -> 
( A  =  B  <-> 
C  =  D ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 103  DECID wdc 780    = wceq 1289    =/= wne 2255
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-in2 580  ax-io 665
This theorem depends on definitions:  df-bi 115  df-dc 781  df-ne 2256
This theorem is referenced by:  nebidc  2335
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