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Theorem nfxnegd 45391
Description: Deduction version of nfxneg 45411. (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 13152 . 2 -𝑒𝐴 = if(𝐴 = +∞, -∞, if(𝐴 = -∞, +∞, -𝐴))
2 nfxnegd.1 . . . 4 (𝜑𝑥𝐴)
3 nfcvd 2904 . . . 4 (𝜑𝑥+∞)
42, 3nfeqd 2914 . . 3 (𝜑 → Ⅎ𝑥 𝐴 = +∞)
5 nfcvd 2904 . . 3 (𝜑𝑥-∞)
62, 5nfeqd 2914 . . . 4 (𝜑 → Ⅎ𝑥 𝐴 = -∞)
72nfnegd 11501 . . . 4 (𝜑𝑥-𝐴)
86, 3, 7nfifd 4560 . . 3 (𝜑𝑥if(𝐴 = -∞, +∞, -𝐴))
94, 5, 8nfifd 4560 . 2 (𝜑𝑥if(𝐴 = +∞, -∞, if(𝐴 = -∞, +∞, -𝐴)))
101, 9nfcxfrd 2902 1 (𝜑𝑥-𝑒𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wnfc 2888  ifcif 4531  +∞cpnf 11290  -∞cmnf 11291  -cneg 11491  -𝑒cxne 13149
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-iota 6516  df-fv 6571  df-ov 7434  df-neg 11493  df-xneg 13152
This theorem is referenced by:  nfxneg  45411
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