Theorem nfnegd 11199

Description: Deduction version of nfneg 11200. (Contributed by NM, 29-Feb-2008.) (Revised by Mario Carneiro, 15-Oct-2016.)

Hypothesis

Ref	Expression
nfnegd.1	⊢ (𝜑 → Ⅎ𝑥𝐴)

Assertion

Ref	Expression
nfnegd	⊢ (𝜑 → Ⅎ𝑥-𝐴)

Proof of Theorem nfnegd

Step	Hyp	Ref	Expression
1		df-neg 11191	. 2 ⊢ -𝐴 = (0 − 𝐴)
2		nfcvd 2909	. . 3 ⊢ (𝜑 → Ⅎ𝑥0)
3		nfcvd 2909	. . 3 ⊢ (𝜑 → Ⅎ𝑥 − )
4		nfnegd.1	. . 3 ⊢ (𝜑 → Ⅎ𝑥𝐴)
5	2, 3, 4	nfovd 7297	. 2 ⊢ (𝜑 → Ⅎ𝑥(0 − 𝐴))
6	1, 5	nfcxfrd 2907	1 ⊢ (𝜑 → Ⅎ𝑥-𝐴)

Syntax hints:   → wi 4  Ⅎwnfc 2888  (class class class)co 7268  0cc0 10855   − cmin 11188  -cneg 11189

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-ext 2710

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-nf 1790  df-sb 2071  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ral 3070  df-rex 3071  df-rab 3074  df-v 3432  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-nul 4262  df-if 4465  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4845  df-br 5079  df-iota 6388  df-fv 6438  df-ov 7271  df-neg 11191

This theorem is referenced by:  nfneg  11200  nfxnegd  42935