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Theorem negeq 7654
Description: Equality theorem for negatives. (Contributed by NM, 10-Feb-1995.)
Assertion
Ref Expression
negeq  |-  ( A  =  B  ->  -u A  =  -u B )

Proof of Theorem negeq
StepHypRef Expression
1 oveq2 5642 . 2  |-  ( A  =  B  ->  (
0  -  A )  =  ( 0  -  B ) )
2 df-neg 7635 . 2  |-  -u A  =  ( 0  -  A )
3 df-neg 7635 . 2  |-  -u B  =  ( 0  -  B )
41, 2, 33eqtr4g 2145 1  |-  ( A  =  B  ->  -u A  =  -u B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1289  (class class class)co 5634   0cc0 7329    - cmin 7632   -ucneg 7633
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-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-rex 2365  df-v 2621  df-un 3001  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-br 3838  df-iota 4967  df-fv 5010  df-ov 5637  df-neg 7635
This theorem is referenced by:  negeqi  7655  negeqd  7656  neg11  7712  negf1o  7839  recexre  8031  negiso  8388  elz  8722  znegcl  8751  zaddcllemneg  8759  elz2  8788  zindd  8834  infrenegsupex  9051  supinfneg  9052  infsupneg  9053  supminfex  9054  ublbneg  9067  eqreznegel  9068  negm  9069  qnegcl  9090  xnegeq  9258  ceilqval  9678  exp3val  9922  expnegap0  9928  m1expcl2  9942  negfi  10623  dvdsnegb  10906  infssuzex  11038  infssuzcldc  11040  lcmneg  11149  znnen  11304  ex-ceil  11310
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