Theorem nfnegd 8457

Description: Deduction version of nfneg 8458. (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

Step	Hyp	Ref	Expression
1		df-neg 8435	. 2 |- -u A = ( 0 - A )
2		nfcvd 2385	. . 3 |- ( ph -> F/_ x 0 )
3		nfcvd 2385	. . 3 |- ( ph -> F/_ x - )
4		nfnegd.1	. . 3 |- ( ph -> F/_ x A )
5	2, 3, 4	nfovd 6070	. 2 |- ( ph -> F/_ x ( 0 - A ) )
6	1, 5	nfcxfrd 2382	1 |- ( ph -> F/_ x -u A )

Syntax hints:   -> wi 4   F/_wnfc 2371  (class class class)co 6041   0cc0 8115   - cmin 8432   -ucneg 8433

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-rex 2526  df-v 2814  df-un 3214  df-sn 3688  df-pr 3689  df-op 3691  df-uni 3908  df-br 4103  df-iota 5303  df-fv 5351  df-ov 6044  df-neg 8435

This theorem is referenced by:  nfneg  8458