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Mirrors > Home > MPE Home > Th. List > nfnegd | Structured version Visualization version GIF version |
Description: Deduction version of nfneg 10156. (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 10148 | . 2 ⊢ -𝐴 = (0 − 𝐴) | |
2 | nfcvd 2752 | . . 3 ⊢ (𝜑 → Ⅎ𝑥0) | |
3 | nfcvd 2752 | . . 3 ⊢ (𝜑 → Ⅎ𝑥 − ) | |
4 | nfnegd.1 | . . 3 ⊢ (𝜑 → Ⅎ𝑥𝐴) | |
5 | 2, 3, 4 | nfovd 6574 | . 2 ⊢ (𝜑 → Ⅎ𝑥(0 − 𝐴)) |
6 | 1, 5 | nfcxfrd 2750 | 1 ⊢ (𝜑 → Ⅎ𝑥-𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 Ⅎwnfc 2738 (class class class)co 6549 0cc0 9815 − cmin 10145 -cneg 10146 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-iota 5768 df-fv 5812 df-ov 6552 df-neg 10148 |
This theorem is referenced by: nfneg 10156 |
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