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Mirrors > Home > MPE Home > Th. List > nfnegd | Structured version Visualization version GIF version |
Description: Deduction version of nfneg 11502. (Contributed by NM, 29-Feb-2008.) (Revised by Mario Carneiro, 15-Oct-2016.) |
Ref | Expression |
---|---|
nfnegd.1 | ⊢ (𝜑 → Ⅎ𝑥𝐴) |
Ref | Expression |
---|---|
nfnegd | ⊢ (𝜑 → Ⅎ𝑥-𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-neg 11493 | . 2 ⊢ -𝐴 = (0 − 𝐴) | |
2 | nfcvd 2904 | . . 3 ⊢ (𝜑 → Ⅎ𝑥0) | |
3 | nfcvd 2904 | . . 3 ⊢ (𝜑 → Ⅎ𝑥 − ) | |
4 | nfnegd.1 | . . 3 ⊢ (𝜑 → Ⅎ𝑥𝐴) | |
5 | 2, 3, 4 | nfovd 7460 | . 2 ⊢ (𝜑 → Ⅎ𝑥(0 − 𝐴)) |
6 | 1, 5 | nfcxfrd 2902 | 1 ⊢ (𝜑 → Ⅎ𝑥-𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 Ⅎwnfc 2888 (class class class)co 7431 0cc0 11153 − cmin 11490 -cneg 11491 |
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 |
This theorem is referenced by: nfneg 11502 nfxnegd 45391 |
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