Theorem nfxnegd 46451

Description: Deduction version of nfxneg 46471. (Contributed by Glauco Siliprandi, 2-Jan-2022.)

Hypothesis

Ref	Expression
nfxnegd.1	⊢ (𝜑 → Ⅎ𝑥𝐴)

Assertion

Ref	Expression
nfxnegd	⊢ (𝜑 → Ⅎ𝑥 -e𝐴)

Proof of Theorem nfxnegd

Step	Hyp	Ref	Expression
1		df-xneg 13241	. 2 ⊢ -e𝐴 = if(𝐴 = +∞, -∞, if(𝐴 = -∞, +∞, -𝐴))
2		nfxnegd.1	. . . 4 ⊢ (𝜑 → Ⅎ𝑥𝐴)
3		nfcvd 2923	. . . 4 ⊢ (𝜑 → Ⅎ𝑥+∞)
4	2, 3	nfeqd 2932	. . 3 ⊢ (𝜑 → Ⅎ𝑥 𝐴 = +∞)
5		nfcvd 2923	. . 3 ⊢ (𝜑 → Ⅎ𝑥-∞)
6	2, 5	nfeqd 2932	. . . 4 ⊢ (𝜑 → Ⅎ𝑥 𝐴 = -∞)
7	2	nfnegd 11552	. . . 4 ⊢ (𝜑 → Ⅎ𝑥 -𝐴)
8	6, 3, 7	nfifd 4511	. . 3 ⊢ (𝜑 → Ⅎ𝑥if(𝐴 = -∞, +∞, -𝐴))
9	4, 5, 8	nfifd 4511	. 2 ⊢ (𝜑 → Ⅎ𝑥if(𝐴 = +∞, -∞, if(𝐴 = -∞, +∞, -𝐴)))
10	1, 9	nfcxfrd 2921	1 ⊢ (𝜑 → Ⅎ𝑥 -e𝐴)

This proof depends on syntax axioms:   → wi 4   = wceq 1569  Ⅎwnfc 2907  ifcif 4481  +∞cpnf 11340  -∞cmnf 11341   -cneg 11542   -_ecxne 13238

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3899  df-un 3901  df-ss 3914  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-iota 6494  df-fv 6546  df-ov 7423  df-neg 11544  df-xneg 13241

This theorem is used by:  nfxneg  46471