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| Mirrors > Home > MPE Home > Th. List > Mathboxes > nfxnegd | Structured version Visualization version GIF version | ||
| Description: Deduction version of nfxneg 45472. (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 13154 | . 2 ⊢ -𝑒𝐴 = if(𝐴 = +∞, -∞, if(𝐴 = -∞, +∞, -𝐴)) | |
| 2 | nfxnegd.1 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥𝐴) | |
| 3 | nfcvd 2906 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥+∞) | |
| 4 | 2, 3 | nfeqd 2916 | . . 3 ⊢ (𝜑 → Ⅎ𝑥 𝐴 = +∞) | 
| 5 | nfcvd 2906 | . . 3 ⊢ (𝜑 → Ⅎ𝑥-∞) | |
| 6 | 2, 5 | nfeqd 2916 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥 𝐴 = -∞) | 
| 7 | 2 | nfnegd 11503 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥-𝐴) | 
| 8 | 6, 3, 7 | nfifd 4555 | . . 3 ⊢ (𝜑 → Ⅎ𝑥if(𝐴 = -∞, +∞, -𝐴)) | 
| 9 | 4, 5, 8 | nfifd 4555 | . 2 ⊢ (𝜑 → Ⅎ𝑥if(𝐴 = +∞, -∞, if(𝐴 = -∞, +∞, -𝐴))) | 
| 10 | 1, 9 | nfcxfrd 2904 | 1 ⊢ (𝜑 → Ⅎ𝑥-𝑒𝐴) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1540 Ⅎwnfc 2890 ifcif 4525 +∞cpnf 11292 -∞cmnf 11293 -cneg 11493 -𝑒cxne 13151 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-iota 6514 df-fv 6569 df-ov 7434 df-neg 11495 df-xneg 13154 | 
| This theorem is referenced by: nfxneg 45472 | 
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