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

Theorem negeqd 7206
 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 7204 . 2 (𝐴 = 𝐵 → -𝐴 = -𝐵)
31, 2syl 14 1 (𝜑 → -𝐴 = -𝐵)
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1243  -cneg 7183 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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rex 2312  df-v 2559  df-un 2922  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-iota 4867  df-fv 4910  df-ov 5515  df-neg 7185 This theorem is referenced by:  negdi  7268  mulneg2  7393  mulm1  7397  mulreim  7595  apneg  7602  divnegap  7683  div2negap  7711  recgt0  7816  ceilqval  9148  ceilid  9157  monoord2  9236  reneg  9468  imneg  9476  cjcj  9483  cjneg  9490  ex-ceil  9896
 Copyright terms: Public domain W3C validator