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Theorem necon4ddc 2321
Description: Contrapositive inference for inequality. (Contributed by Jim Kingdon, 17-May-2018.)
Hypothesis
Ref Expression
necon4ddc.1  |-  ( ph  ->  (DECID  A  =  B  -> 
( A  =/=  B  ->  C  =/=  D ) ) )
Assertion
Ref Expression
necon4ddc  |-  ( ph  ->  (DECID  A  =  B  -> 
( C  =  D  ->  A  =  B ) ) )

Proof of Theorem necon4ddc
StepHypRef Expression
1 necon4ddc.1 . . 3  |-  ( ph  ->  (DECID  A  =  B  -> 
( A  =/=  B  ->  C  =/=  D ) ) )
2 df-ne 2250 . . . 4  |-  ( A  =/=  B  <->  -.  A  =  B )
3 df-ne 2250 . . . 4  |-  ( C  =/=  D  <->  -.  C  =  D )
42, 3imbi12i 237 . . 3  |-  ( ( A  =/=  B  ->  C  =/=  D )  <->  ( -.  A  =  B  ->  -.  C  =  D ) )
51, 4syl6ib 159 . 2  |-  ( ph  ->  (DECID  A  =  B  -> 
( -.  A  =  B  ->  -.  C  =  D ) ) )
6 condc 783 . 2  |-  (DECID  A  =  B  ->  ( ( -.  A  =  B  ->  -.  C  =  D )  ->  ( C  =  D  ->  A  =  B ) ) )
75, 6sylcom 28 1  |-  ( ph  ->  (DECID  A  =  B  -> 
( C  =  D  ->  A  =  B ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4  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-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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