Theorem nfxnegd 45435

Description: Deduction version of nfxneg 45455. (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 13133	. 2 ⊢ -𝑒𝐴 = if(𝐴 = +∞, -∞, if(𝐴 = -∞, +∞, -𝐴))
2		nfxnegd.1	. . . 4 ⊢ (𝜑 → Ⅎ𝑥𝐴)
3		nfcvd 2900	. . . 4 ⊢ (𝜑 → Ⅎ𝑥+∞)
4	2, 3	nfeqd 2910	. . 3 ⊢ (𝜑 → Ⅎ𝑥 𝐴 = +∞)
5		nfcvd 2900	. . 3 ⊢ (𝜑 → Ⅎ𝑥-∞)
6	2, 5	nfeqd 2910	. . . 4 ⊢ (𝜑 → Ⅎ𝑥 𝐴 = -∞)
7	2	nfnegd 11482	. . . 4 ⊢ (𝜑 → Ⅎ𝑥-𝐴)
8	6, 3, 7	nfifd 4535	. . 3 ⊢ (𝜑 → Ⅎ𝑥if(𝐴 = -∞, +∞, -𝐴))
9	4, 5, 8	nfifd 4535	. 2 ⊢ (𝜑 → Ⅎ𝑥if(𝐴 = +∞, -∞, if(𝐴 = -∞, +∞, -𝐴)))
10	1, 9	nfcxfrd 2898	1 ⊢ (𝜑 → Ⅎ𝑥-𝑒𝐴)

Syntax hints:   → wi 4   = wceq 1540  Ⅎwnfc 2884  ifcif 4505  +∞cpnf 11271  -∞cmnf 11272  -cneg 11472  -_𝑒cxne 13130

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-br 5125  df-iota 6489  df-fv 6544  df-ov 7413  df-neg 11474  df-xneg 13133

This theorem is referenced by:  nfxneg  45455