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Theorem negned 9408
Description: If two complex numbers are unequal, so are their negatives. Contrapositive of neg11d 9423. (Contributed by David Moews, 28-Feb-2017.)
Hypotheses
Ref Expression
negidd.1  |-  ( ph  ->  A  e.  CC )
negned.2  |-  ( ph  ->  B  e.  CC )
negned.3  |-  ( ph  ->  A  =/=  B )
Assertion
Ref Expression
negned  |-  ( ph  -> 
-u A  =/=  -u B
)

Proof of Theorem negned
StepHypRef Expression
1 negned.3 . 2  |-  ( ph  ->  A  =/=  B )
2 negidd.1 . . . 4  |-  ( ph  ->  A  e.  CC )
3 negned.2 . . . 4  |-  ( ph  ->  B  e.  CC )
42, 3neg11ad 9407 . . 3  |-  ( ph  ->  ( -u A  = 
-u B  <->  A  =  B ) )
54necon3bid 2636 . 2  |-  ( ph  ->  ( -u A  =/=  -u B  <->  A  =/=  B
) )
61, 5mpbird 224 1  |-  ( ph  -> 
-u A  =/=  -u B
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1725    =/= wne 2599   CCcc 8988   -ucneg 9292
This theorem is referenced by:  angpieqvdlem  20669
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-po 4503  df-so 4504  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-riota 6549  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-pnf 9122  df-mnf 9123  df-ltxr 9125  df-sub 9293  df-neg 9294
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