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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > nfxnegd | Structured version Visualization version GIF version |
Description: Deduction version of nfxneg 44845. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
Ref | Expression |
---|---|
nfxnegd.1 | ⊢ (𝜑 → Ⅎ𝑥𝐴) |
Ref | Expression |
---|---|
nfxnegd | ⊢ (𝜑 → Ⅎ𝑥-𝑒𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-xneg 13130 | . 2 ⊢ -𝑒𝐴 = if(𝐴 = +∞, -∞, if(𝐴 = -∞, +∞, -𝐴)) | |
2 | nfxnegd.1 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥𝐴) | |
3 | nfcvd 2899 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥+∞) | |
4 | 2, 3 | nfeqd 2909 | . . 3 ⊢ (𝜑 → Ⅎ𝑥 𝐴 = +∞) |
5 | nfcvd 2899 | . . 3 ⊢ (𝜑 → Ⅎ𝑥-∞) | |
6 | 2, 5 | nfeqd 2909 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥 𝐴 = -∞) |
7 | 2 | nfnegd 11491 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥-𝐴) |
8 | 6, 3, 7 | nfifd 4559 | . . 3 ⊢ (𝜑 → Ⅎ𝑥if(𝐴 = -∞, +∞, -𝐴)) |
9 | 4, 5, 8 | nfifd 4559 | . 2 ⊢ (𝜑 → Ⅎ𝑥if(𝐴 = +∞, -∞, if(𝐴 = -∞, +∞, -𝐴))) |
10 | 1, 9 | nfcxfrd 2897 | 1 ⊢ (𝜑 → Ⅎ𝑥-𝑒𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 Ⅎwnfc 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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