Theorem nfnegd 8222

Description: Deduction version of nfneg 8223. (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 8200	. 2 ⊢ -𝐴 = (0 − 𝐴)
2		nfcvd 2340	. . 3 ⊢ (𝜑 → Ⅎ𝑥0)
3		nfcvd 2340	. . 3 ⊢ (𝜑 → Ⅎ𝑥 − )
4		nfnegd.1	. . 3 ⊢ (𝜑 → Ⅎ𝑥𝐴)
5	2, 3, 4	nfovd 5951	. 2 ⊢ (𝜑 → Ⅎ𝑥(0 − 𝐴))
6	1, 5	nfcxfrd 2337	1 ⊢ (𝜑 → Ⅎ𝑥-𝐴)

Syntax hints:   → wi 4  Ⅎwnfc 2326  (class class class)co 5922  0cc0 7879   − cmin 8197  -cneg 8198

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-rex 2481  df-v 2765  df-un 3161  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-iota 5219  df-fv 5266  df-ov 5925  df-neg 8200

This theorem is referenced by:  nfneg  8223