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Theorem negcon1 7326
Description: Negative contraposition law. (Contributed by NM, 9-May-2004.)
Assertion
Ref Expression
negcon1  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( -u A  =  B  <->  -u B  =  A ) )

Proof of Theorem negcon1
StepHypRef Expression
1 negcl 7274 . . . 4  |-  ( A  e.  CC  ->  -u A  e.  CC )
2 neg11 7325 . . . 4  |-  ( (
-u A  e.  CC  /\  B  e.  CC )  ->  ( -u -u A  =  -u B  <->  -u A  =  B ) )
31, 2sylan 271 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( -u -u A  =  -u B  <->  -u A  =  B ) )
4 negneg 7324 . . . . 5  |-  ( A  e.  CC  ->  -u -u A  =  A )
54adantr 265 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  -> 
-u -u A  =  A )
65eqeq1d 2064 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( -u -u A  =  -u B  <->  A  =  -u B ) )
73, 6bitr3d 183 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( -u A  =  B  <->  A  =  -u B
) )
8 eqcom 2058 . 2  |-  ( A  =  -u B  <->  -u B  =  A )
97, 8syl6bb 189 1  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( -u A  =  B  <->  -u B  =  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    <-> wb 102    = wceq 1259    e. wcel 1409   CCcc 6945   -ucneg 7246
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  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972  ax-setind 4290  ax-resscn 7034  ax-1cn 7035  ax-icn 7037  ax-addcl 7038  ax-addrcl 7039  ax-mulcl 7040  ax-addcom 7042  ax-addass 7044  ax-distr 7046  ax-i2m1 7047  ax-0id 7050  ax-rnegex 7051  ax-cnre 7053
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2788  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-br 3793  df-opab 3847  df-id 4058  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-iota 4895  df-fun 4932  df-fv 4938  df-riota 5496  df-ov 5543  df-oprab 5544  df-mpt2 5545  df-sub 7247  df-neg 7248
This theorem is referenced by:  negcon2  7327  negcon1i  7356  negcon1d  7379  elznn0  8317
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