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Theorem nfxnegd 44825
Description: Deduction version of nfxneg 44845. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
Hypothesis
Ref Expression
nfxnegd.1 (𝜑𝑥𝐴)
Assertion
Ref Expression
nfxnegd (𝜑𝑥-𝑒𝐴)

Proof of Theorem nfxnegd
StepHypRef Expression
1 df-xneg 13130 . 2 -𝑒𝐴 = if(𝐴 = +∞, -∞, if(𝐴 = -∞, +∞, -𝐴))
2 nfxnegd.1 . . . 4 (𝜑𝑥𝐴)
3 nfcvd 2899 . . . 4 (𝜑𝑥+∞)
42, 3nfeqd 2909 . . 3 (𝜑 → Ⅎ𝑥 𝐴 = +∞)
5 nfcvd 2899 . . 3 (𝜑𝑥-∞)
62, 5nfeqd 2909 . . . 4 (𝜑 → Ⅎ𝑥 𝐴 = -∞)
72nfnegd 11491 . . . 4 (𝜑𝑥-𝐴)
86, 3, 7nfifd 4559 . . 3 (𝜑𝑥if(𝐴 = -∞, +∞, -𝐴))
94, 5, 8nfifd 4559 . 2 (𝜑𝑥if(𝐴 = +∞, -∞, if(𝐴 = -∞, +∞, -𝐴)))
101, 9nfcxfrd 2897 1 (𝜑𝑥-𝑒𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wnfc 2878  ifcif 4530  +∞cpnf 11281  -∞cmnf 11282  -cneg 11481  -𝑒cxne 13127
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2698
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-br 5151  df-iota 6503  df-fv 6559  df-ov 7427  df-neg 11483  df-xneg 13130
This theorem is referenced by:  nfxneg  44845
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