Theorem negeq 7001

Description: Equality theorem for negatives. (Contributed by NM, 10-Feb-1995.)

Assertion

Ref	Expression
negeq	⊢ (A = B → -A = -B)

Proof of Theorem negeq

Step	Hyp	Ref	Expression
1		oveq2 5463	. 2 ⊢ (A = B → (0 − A) = (0 − B))
2		df-neg 6982	. 2 ⊢ -A = (0 − A)
3		df-neg 6982	. 2 ⊢ -B = (0 − B)
4	1, 2, 3	3eqtr4g 2094	1 ⊢ (A = B → -A = -B)

Syntax hints:   → wi 4   = wceq 1242  (class class class)co 5455  0cc0 6711   − cmin 6979  -cneg 6980

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rex 2306  df-v 2553  df-un 2916  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-iota 4810  df-fv 4853  df-ov 5458  df-neg 6982

This theorem is referenced by:  negeqi  7002  negeqd  7003  neg11  7058  recexre  7362  elz  8023  znegcl  8052  zaddcllemneg  8060  elz2  8088  zindd  8132  ublbneg  8324  eqreznegel  8325  negm  8326  qnegcl  8347  xnegeq  8510  expival  8911  expnegap0  8917  m1expcl2  8931