Theorem nfxnegd 46593

Description: Deduction version of nfxneg 46613. (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 13254	. 2 ⊢ -e𝐴 = if(𝐴 = +∞, -∞, if(𝐴 = -∞, +∞, -𝐴))
2		nfxnegd.1	. . . 4 ⊢ (𝜑 → Ⅎ𝑥𝐴)
3		nfcvd 2927	. . . 4 ⊢ (𝜑 → Ⅎ𝑥+∞)
4	2, 3	nfeqd 2936	. . 3 ⊢ (𝜑 → Ⅎ𝑥 𝐴 = +∞)
5		nfcvd 2927	. . 3 ⊢ (𝜑 → Ⅎ𝑥-∞)
6	2, 5	nfeqd 2936	. . . 4 ⊢ (𝜑 → Ⅎ𝑥 𝐴 = -∞)
7	2	nfnegd 11565	. . . 4 ⊢ (𝜑 → Ⅎ𝑥 -𝐴)
8	6, 3, 7	nfifd 4515	. . 3 ⊢ (𝜑 → Ⅎ𝑥if(𝐴 = -∞, +∞, -𝐴))
9	4, 5, 8	nfifd 4515	. 2 ⊢ (𝜑 → Ⅎ𝑥if(𝐴 = +∞, -∞, if(𝐴 = -∞, +∞, -𝐴)))
10	1, 9	nfcxfrd 2925	1 ⊢ (𝜑 → Ⅎ𝑥 -e𝐴)

This proof depends on syntax axioms:   → wi 4   = wceq 1569  Ⅎwnfc 2911  ifcif 4485  +∞cpnf 11353  -∞cmnf 11354   -cneg 11555   -_ecxne 13251

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 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735

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 2101  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-iota 6501  df-fv 6555  df-ov 7433  df-neg 11557  df-xneg 13254

This theorem is used by:  nfxneg  46613