ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  negeqd GIF version

Theorem negeqd 7269
Description: Equality deduction for negatives. (Contributed by NM, 14-May-1999.)
Hypothesis
Ref Expression
negeqd.1 (𝜑𝐴 = 𝐵)
Assertion
Ref Expression
negeqd (𝜑 → -𝐴 = -𝐵)

Proof of Theorem negeqd
StepHypRef Expression
1 negeqd.1 . 2 (𝜑𝐴 = 𝐵)
2 negeq 7267 . 2 (𝐴 = 𝐵 → -𝐴 = -𝐵)
31, 2syl 14 1 (𝜑 → -𝐴 = -𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1259  -cneg 7246
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 2950  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-br 3793  df-iota 4895  df-fv 4938  df-ov 5543  df-neg 7248
This theorem is referenced by:  negdi  7331  mulneg2  7465  mulm1  7469  mulreim  7669  apneg  7676  divnegap  7757  div2negap  7786  recgt0  7891  ceilqval  9256  ceilid  9265  modqcyc2  9310  monoord2  9400  reneg  9696  imneg  9704  cjcj  9711  cjneg  9718  odd2np1  10184  oexpneg  10188  ex-ceil  10280
  Copyright terms: Public domain W3C validator