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Mirrors > Home > MPE Home > Th. List > Mathboxes > nfxnegd | Structured version Visualization version GIF version |
Description: Deduction version of nfxneg 39691. (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 11946 | . 2 ⊢ -𝑒𝐴 = if(𝐴 = +∞, -∞, if(𝐴 = -∞, +∞, -𝐴)) | |
2 | nfxnegd.1 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥𝐴) | |
3 | nfcvd 2765 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥+∞) | |
4 | 2, 3 | nfeqd 2772 | . . 3 ⊢ (𝜑 → Ⅎ𝑥 𝐴 = +∞) |
5 | nfcvd 2765 | . . 3 ⊢ (𝜑 → Ⅎ𝑥-∞) | |
6 | 2, 5 | nfeqd 2772 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥 𝐴 = -∞) |
7 | 2 | nfnegd 10276 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥-𝐴) |
8 | 6, 3, 7 | nfifd 4114 | . . 3 ⊢ (𝜑 → Ⅎ𝑥if(𝐴 = -∞, +∞, -𝐴)) |
9 | 4, 5, 8 | nfifd 4114 | . 2 ⊢ (𝜑 → Ⅎ𝑥if(𝐴 = +∞, -∞, if(𝐴 = -∞, +∞, -𝐴))) |
10 | 1, 9 | nfcxfrd 2763 | 1 ⊢ (𝜑 → Ⅎ𝑥-𝑒𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 Ⅎwnfc 2751 ifcif 4086 +∞cpnf 10071 -∞cmnf 10072 -cneg 10267 -𝑒cxne 11943 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-iota 5851 df-fv 5896 df-ov 6653 df-neg 10269 df-xneg 11946 |
This theorem is referenced by: nfxneg 39691 |
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