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Mirrors > Home > MPE Home > Th. List > Mathboxes > nfxnegd | Structured version Visualization version GIF version |
Description: Deduction version of nfxneg 45411. (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 13152 | . 2 ⊢ -𝑒𝐴 = if(𝐴 = +∞, -∞, if(𝐴 = -∞, +∞, -𝐴)) | |
2 | nfxnegd.1 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥𝐴) | |
3 | nfcvd 2904 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥+∞) | |
4 | 2, 3 | nfeqd 2914 | . . 3 ⊢ (𝜑 → Ⅎ𝑥 𝐴 = +∞) |
5 | nfcvd 2904 | . . 3 ⊢ (𝜑 → Ⅎ𝑥-∞) | |
6 | 2, 5 | nfeqd 2914 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥 𝐴 = -∞) |
7 | 2 | nfnegd 11501 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥-𝐴) |
8 | 6, 3, 7 | nfifd 4560 | . . 3 ⊢ (𝜑 → Ⅎ𝑥if(𝐴 = -∞, +∞, -𝐴)) |
9 | 4, 5, 8 | nfifd 4560 | . 2 ⊢ (𝜑 → Ⅎ𝑥if(𝐴 = +∞, -∞, if(𝐴 = -∞, +∞, -𝐴))) |
10 | 1, 9 | nfcxfrd 2902 | 1 ⊢ (𝜑 → Ⅎ𝑥-𝑒𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 Ⅎwnfc 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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