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Theorem nfxnegd 40584
Description: Deduction version of nfxneg 40606. (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 12261 . 2 -𝑒𝐴 = if(𝐴 = +∞, -∞, if(𝐴 = -∞, +∞, -𝐴))
2 nfxnegd.1 . . . 4 (𝜑𝑥𝐴)
3 nfcvd 2935 . . . 4 (𝜑𝑥+∞)
42, 3nfeqd 2942 . . 3 (𝜑 → Ⅎ𝑥 𝐴 = +∞)
5 nfcvd 2935 . . 3 (𝜑𝑥-∞)
62, 5nfeqd 2942 . . . 4 (𝜑 → Ⅎ𝑥 𝐴 = -∞)
72nfnegd 10619 . . . 4 (𝜑𝑥-𝐴)
86, 3, 7nfifd 4335 . . 3 (𝜑𝑥if(𝐴 = -∞, +∞, -𝐴))
94, 5, 8nfifd 4335 . 2 (𝜑𝑥if(𝐴 = +∞, -∞, if(𝐴 = -∞, +∞, -𝐴)))
101, 9nfcxfrd 2933 1 (𝜑𝑥-𝑒𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1601  wnfc 2919  ifcif 4307  +∞cpnf 10410  -∞cmnf 10411  -cneg 10609  -𝑒cxne 12258
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4674  df-br 4889  df-iota 6101  df-fv 6145  df-ov 6927  df-neg 10611  df-xneg 12261
This theorem is referenced by:  nfxneg  40606
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