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Mirrors > Home > MPE Home > Th. List > nfnegd | Structured version Visualization version GIF version |
Description: Deduction version of nfneg 10882. (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 10873 | . 2 ⊢ -𝐴 = (0 − 𝐴) | |
2 | nfcvd 2978 | . . 3 ⊢ (𝜑 → Ⅎ𝑥0) | |
3 | nfcvd 2978 | . . 3 ⊢ (𝜑 → Ⅎ𝑥 − ) | |
4 | nfnegd.1 | . . 3 ⊢ (𝜑 → Ⅎ𝑥𝐴) | |
5 | 2, 3, 4 | nfovd 7185 | . 2 ⊢ (𝜑 → Ⅎ𝑥(0 − 𝐴)) |
6 | 1, 5 | nfcxfrd 2976 | 1 ⊢ (𝜑 → Ⅎ𝑥-𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 Ⅎwnfc 2961 (class class class)co 7156 0cc0 10537 − cmin 10870 -cneg 10871 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 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 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-iota 6314 df-fv 6363 df-ov 7159 df-neg 10873 |
This theorem is referenced by: nfneg 10882 nfxnegd 41735 |
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