Theorem nfnegd 8127

Description: Deduction version of nfneg 8128. (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 8105	. 2 ⊢ -𝐴 = (0 − 𝐴)
2		nfcvd 2318	. . 3 ⊢ (𝜑 → Ⅎ𝑥0)
3		nfcvd 2318	. . 3 ⊢ (𝜑 → Ⅎ𝑥 − )
4		nfnegd.1	. . 3 ⊢ (𝜑 → Ⅎ𝑥𝐴)
5	2, 3, 4	nfovd 5894	. 2 ⊢ (𝜑 → Ⅎ𝑥(0 − 𝐴))
6	1, 5	nfcxfrd 2315	1 ⊢ (𝜑 → Ⅎ𝑥-𝐴)

Syntax hints:   → wi 4  Ⅎwnfc 2304  (class class class)co 5865  0cc0 7786   − cmin 8102  -cneg 8103

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 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-rex 2459  df-v 2737  df-un 3131  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-br 3999  df-iota 5170  df-fv 5216  df-ov 5868  df-neg 8105

This theorem is referenced by:  nfneg  8128