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Theorem negeq 7266
Description: Equality theorem for negatives. (Contributed by NM, 10-Feb-1995.)
Assertion
Ref Expression
negeq (𝐴 = 𝐵 → -𝐴 = -𝐵)

Proof of Theorem negeq
StepHypRef Expression
1 oveq2 5547 . 2 (𝐴 = 𝐵 → (0 − 𝐴) = (0 − 𝐵))
2 df-neg 7247 . 2 -𝐴 = (0 − 𝐴)
3 df-neg 7247 . 2 -𝐵 = (0 − 𝐵)
41, 2, 33eqtr4g 2113 1 (𝐴 = 𝐵 → -𝐴 = -𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1259  (class class class)co 5539  0cc0 6946  cmin 7244  -cneg 7245
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-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-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-rex 2329  df-v 2576  df-un 2949  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-br 3792  df-iota 4894  df-fv 4937  df-ov 5542  df-neg 7247
This theorem is referenced by:  negeqi  7267  negeqd  7268  neg11  7324  recexre  7642  elz  8303  znegcl  8332  zaddcllemneg  8340  elz2  8369  zindd  8414  ublbneg  8644  eqreznegel  8645  negm  8646  qnegcl  8667  xnegeq  8840  ceilqval  9255  expival  9421  expnegap0  9427  m1expcl2  9441  dvdsnegb  10124  ex-ceil  10259
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