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Mirrors > Home > MPE Home > Th. List > nfnegd | Structured version Visualization version GIF version |
Description: Deduction version of nfneg 10870. (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 10861 | . 2 ⊢ -𝐴 = (0 − 𝐴) | |
2 | nfcvd 2975 | . . 3 ⊢ (𝜑 → Ⅎ𝑥0) | |
3 | nfcvd 2975 | . . 3 ⊢ (𝜑 → Ⅎ𝑥 − ) | |
4 | nfnegd.1 | . . 3 ⊢ (𝜑 → Ⅎ𝑥𝐴) | |
5 | 2, 3, 4 | nfovd 7174 | . 2 ⊢ (𝜑 → Ⅎ𝑥(0 − 𝐴)) |
6 | 1, 5 | nfcxfrd 2973 | 1 ⊢ (𝜑 → Ⅎ𝑥-𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 Ⅎwnfc 2958 (class class class)co 7145 0cc0 10525 − cmin 10858 -cneg 10859 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-iota 6307 df-fv 6356 df-ov 7148 df-neg 10861 |
This theorem is referenced by: nfneg 10870 nfxnegd 41591 |
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