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

Theorem nfnegd 9901
Description: Deduction version of nfneg 9902. (Contributed by NM, 29-Feb-2008.) (Revised by Mario Carneiro, 15-Oct-2016.)
Hypothesis
Ref Expression
nfnegd.1  |-  ( ph  -> 
F/_ x A )
Assertion
Ref Expression
nfnegd  |-  ( ph  -> 
F/_ x -u A
)

Proof of Theorem nfnegd
StepHypRef Expression
1 df-neg 9894 . 2  |-  -u A  =  ( 0  -  A )
2 nfcvd 2604 . . 3  |-  ( ph  -> 
F/_ x 0 )
3 nfcvd 2604 . . 3  |-  ( ph  -> 
F/_ x  -  )
4 nfnegd.1 . . 3  |-  ( ph  -> 
F/_ x A )
52, 3, 4nfovd 6345 . 2  |-  ( ph  -> 
F/_ x ( 0  -  A ) )
61, 5nfcxfrd 2602 1  |-  ( ph  -> 
F/_ x -u A
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   F/_wnfc 2590  (class class class)co 6320   0cc0 9570    - cmin 9891   -ucneg 9892
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ral 2754  df-rex 2755  df-rab 2758  df-v 3059  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4213  df-br 4419  df-iota 5569  df-fv 5613  df-ov 6323  df-neg 9894
This theorem is referenced by:  nfneg  9902
  Copyright terms: Public domain W3C validator