MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nfnegd Structured version   Visualization version   GIF version

Theorem nfnegd 11501
Description: Deduction version of nfneg 11502. (Contributed by NM, 29-Feb-2008.) (Revised by Mario Carneiro, 15-Oct-2016.)
Hypothesis
Ref Expression
nfnegd.1 (𝜑𝑥𝐴)
Assertion
Ref Expression
nfnegd (𝜑𝑥-𝐴)

Proof of Theorem nfnegd
StepHypRef Expression
1 df-neg 11493 . 2 -𝐴 = (0 − 𝐴)
2 nfcvd 2904 . . 3 (𝜑𝑥0)
3 nfcvd 2904 . . 3 (𝜑𝑥 − )
4 nfnegd.1 . . 3 (𝜑𝑥𝐴)
52, 3, 4nfovd 7460 . 2 (𝜑𝑥(0 − 𝐴))
61, 5nfcxfrd 2902 1 (𝜑𝑥-𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wnfc 2888  (class class class)co 7431  0cc0 11153  cmin 11490  -cneg 11491
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-iota 6516  df-fv 6571  df-ov 7434  df-neg 11493
This theorem is referenced by:  nfneg  11502  nfxnegd  45391
  Copyright terms: Public domain W3C validator