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| Mirrors > Home > MPE Home > Th. List > csbnegg | Structured version Visualization version GIF version | ||
| Description: Move class substitution in and out of the negative of a number. (Contributed by NM, 1-Mar-2008.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) |
| Ref | Expression |
|---|---|
| csbnegg | ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌-𝐵 = -⦋𝐴 / 𝑥⦌𝐵) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | csbov2g 7479 | . 2 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(0 − 𝐵) = (0 − ⦋𝐴 / 𝑥⦌𝐵)) | |
| 2 | df-neg 11495 | . . 3 ⊢ -𝐵 = (0 − 𝐵) | |
| 3 | 2 | csbeq2i 3907 | . 2 ⊢ ⦋𝐴 / 𝑥⦌-𝐵 = ⦋𝐴 / 𝑥⦌(0 − 𝐵) |
| 4 | df-neg 11495 | . 2 ⊢ -⦋𝐴 / 𝑥⦌𝐵 = (0 − ⦋𝐴 / 𝑥⦌𝐵) | |
| 5 | 1, 3, 4 | 3eqtr4g 2802 | 1 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌-𝐵 = -⦋𝐴 / 𝑥⦌𝐵) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ⦋csb 3899 (class class class)co 7431 0cc0 11155 − cmin 11492 -cneg 11493 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-dm 5695 df-iota 6514 df-fv 6569 df-ov 7434 df-neg 11495 |
| This theorem is referenced by: dvfsum2 26075 renegclALT 38964 |
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