Theorem nfxnegd 46450

Description: Deduction version of nfxneg 46470. (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 2924	. . . 4 ⊢ (𝜑 → Ⅎ𝑥+∞)
4	2, 3	nfeqd 2933	. . 3 ⊢ (𝜑 → Ⅎ𝑥 𝐴 = +∞)
5		nfcvd 2924	. . 3 ⊢ (𝜑 → Ⅎ𝑥-∞)
6	2, 5	nfeqd 2933	. . . 4 ⊢ (𝜑 → Ⅎ𝑥 𝐴 = -∞)
7	2	nfnegd 11552	. . . 4 ⊢ (𝜑 → Ⅎ𝑥 -𝐴)
8	6, 3, 7	nfifd 4512	. . 3 ⊢ (𝜑 → Ⅎ𝑥if(𝐴 = -∞, +∞, -𝐴))
9	4, 5, 8	nfifd 4512	. 2 ⊢ (𝜑 → Ⅎ𝑥if(𝐴 = +∞, -∞, if(𝐴 = -∞, +∞, -𝐴)))
10	1, 9	nfcxfrd 2922	1 ⊢ (𝜑 → Ⅎ𝑥 -e𝐴)

This proof depends on syntax axioms:   → wi 4   = wceq 1570  Ⅎwnfc 2908  ifcif 4482  +∞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 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-iota 6494  df-fv 6546  df-ov 7423  df-neg 11544  df-xneg 13241

This theorem is used by:  nfxneg  46470