Theorem negeq 7301

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

Step	Hyp	Ref	Expression
1		oveq2 5540	. 2 |- ( A = B -> ( 0 - A ) = ( 0 - B ) )
2		df-neg 7282	. 2 |- -u A = ( 0 - A )
3		df-neg 7282	. 2 |- -u B = ( 0 - B )
4	1, 2, 3	3eqtr4g 2138	1 |- ( A = B -> -u A = -u B )

Syntax hints:   -> wi 4   = wceq 1284  (class class class)co 5532   0cc0 6981   - cmin 7279   -ucneg 7280

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 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-rex 2354  df-v 2603  df-un 2977  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-iota 4887  df-fv 4930  df-ov 5535  df-neg 7282

This theorem is referenced by:  negeqi  7302  negeqd  7303  neg11  7359  negf1o  7486  recexre  7678  negiso  8033  elz  8353  znegcl  8382  zaddcllemneg  8390  elz2  8419  zindd  8465  infrenegsupex  8682  supinfneg  8683  infsupneg  8684  supminfex  8685  ublbneg  8698  eqreznegel  8699  negm  8700  qnegcl  8721  xnegeq  8894  ceilqval  9308  expival  9478  expnegap0  9484  m1expcl2  9498  negfi  10110  dvdsnegb  10212  infssuzex  10345  infssuzcldc  10347  lcmneg  10456  ex-ceil  10564