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Theorem nfxnegd 40178
Description: Deduction version of nfxneg 40201. (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 12150 . 2 -𝑒𝐴 = if(𝐴 = +∞, -∞, if(𝐴 = -∞, +∞, -𝐴))
2 nfxnegd.1 . . . 4 (𝜑𝑥𝐴)
3 nfcvd 2913 . . . 4 (𝜑𝑥+∞)
42, 3nfeqd 2920 . . 3 (𝜑 → Ⅎ𝑥 𝐴 = +∞)
5 nfcvd 2913 . . 3 (𝜑𝑥-∞)
62, 5nfeqd 2920 . . . 4 (𝜑 → Ⅎ𝑥 𝐴 = -∞)
72nfnegd 10477 . . . 4 (𝜑𝑥-𝐴)
86, 3, 7nfifd 4251 . . 3 (𝜑𝑥if(𝐴 = -∞, +∞, -𝐴))
94, 5, 8nfifd 4251 . 2 (𝜑𝑥if(𝐴 = +∞, -∞, if(𝐴 = -∞, +∞, -𝐴)))
101, 9nfcxfrd 2911 1 (𝜑𝑥-𝑒𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1630  wnfc 2899  ifcif 4223  +∞cpnf 10272  -∞cmnf 10273  -cneg 10468  -𝑒cxne 12147
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-iota 5994  df-fv 6039  df-ov 6795  df-neg 10470  df-xneg 12150
This theorem is referenced by:  nfxneg  40201
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