Theorem nfxnegd 46175

Description: Deduction version of nfxneg 46195. (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 13132	. 2 ⊢ -𝑒𝐴 = if(𝐴 = +∞, -∞, if(𝐴 = -∞, +∞, -𝐴))
2		nfxnegd.1	. . . 4 ⊢ (𝜑 → Ⅎ𝑥𝐴)
3		nfcvd 2926	. . . 4 ⊢ (𝜑 → Ⅎ𝑥+∞)
4	2, 3	nfeqd 2935	. . 3 ⊢ (𝜑 → Ⅎ𝑥 𝐴 = +∞)
5		nfcvd 2926	. . 3 ⊢ (𝜑 → Ⅎ𝑥-∞)
6	2, 5	nfeqd 2935	. . . 4 ⊢ (𝜑 → Ⅎ𝑥 𝐴 = -∞)
7	2	nfnegd 11447	. . . 4 ⊢ (𝜑 → Ⅎ𝑥-𝐴)
8	6, 3, 7	nfifd 4517	. . 3 ⊢ (𝜑 → Ⅎ𝑥if(𝐴 = -∞, +∞, -𝐴))
9	4, 5, 8	nfifd 4517	. 2 ⊢ (𝜑 → Ⅎ𝑥if(𝐴 = +∞, -∞, if(𝐴 = -∞, +∞, -𝐴)))
10	1, 9	nfcxfrd 2924	1 ⊢ (𝜑 → Ⅎ𝑥-𝑒𝐴)

Syntax hints:   → wi 4   = wceq 1570  Ⅎwnfc 2910  ifcif 4487  +∞cpnf 11235  -∞cmnf 11236  -cneg 11437  -_𝑒cxne 13129

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-iota 6492  df-fv 6544  df-ov 7413  df-neg 11439  df-xneg 13132

This theorem is referenced by:  nfxneg  46195