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Mirrors > Home > MPE Home > Th. List > nfnegd | Structured version Visualization version GIF version |
Description: Deduction version of nfneg 11494. (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 11485 | . 2 ⊢ -𝐴 = (0 − 𝐴) | |
2 | nfcvd 2900 | . . 3 ⊢ (𝜑 → Ⅎ𝑥0) | |
3 | nfcvd 2900 | . . 3 ⊢ (𝜑 → Ⅎ𝑥 − ) | |
4 | nfnegd.1 | . . 3 ⊢ (𝜑 → Ⅎ𝑥𝐴) | |
5 | 2, 3, 4 | nfovd 7455 | . 2 ⊢ (𝜑 → Ⅎ𝑥(0 − 𝐴)) |
6 | 1, 5 | nfcxfrd 2898 | 1 ⊢ (𝜑 → Ⅎ𝑥-𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 Ⅎwnfc 2879 (class class class)co 7426 0cc0 11146 − cmin 11482 -cneg 11483 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-iota 6505 df-fv 6561 df-ov 7429 df-neg 11485 |
This theorem is referenced by: nfneg 11494 nfxnegd 44852 |
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