Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > nfxnegd | Structured version Visualization version GIF version |
Description: Deduction version of nfxneg 41735. (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 12506 | . 2 ⊢ -𝑒𝐴 = if(𝐴 = +∞, -∞, if(𝐴 = -∞, +∞, -𝐴)) | |
2 | nfxnegd.1 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥𝐴) | |
3 | nfcvd 2978 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥+∞) | |
4 | 2, 3 | nfeqd 2988 | . . 3 ⊢ (𝜑 → Ⅎ𝑥 𝐴 = +∞) |
5 | nfcvd 2978 | . . 3 ⊢ (𝜑 → Ⅎ𝑥-∞) | |
6 | 2, 5 | nfeqd 2988 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥 𝐴 = -∞) |
7 | 2 | nfnegd 10880 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥-𝐴) |
8 | 6, 3, 7 | nfifd 4494 | . . 3 ⊢ (𝜑 → Ⅎ𝑥if(𝐴 = -∞, +∞, -𝐴)) |
9 | 4, 5, 8 | nfifd 4494 | . 2 ⊢ (𝜑 → Ⅎ𝑥if(𝐴 = +∞, -∞, if(𝐴 = -∞, +∞, -𝐴))) |
10 | 1, 9 | nfcxfrd 2976 | 1 ⊢ (𝜑 → Ⅎ𝑥-𝑒𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 Ⅎwnfc 2961 ifcif 4466 +∞cpnf 10671 -∞cmnf 10672 -cneg 10870 -𝑒cxne 12503 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-br 5066 df-iota 6313 df-fv 6362 df-ov 7158 df-neg 10872 df-xneg 12506 |
This theorem is referenced by: nfxneg 41735 |
Copyright terms: Public domain | W3C validator |