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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 7452 | . 2 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(0 − 𝐵) = (0 − ⦋𝐴 / 𝑥⦌𝐵)) | |
2 | df-neg 11444 | . . 3 ⊢ -𝐵 = (0 − 𝐵) | |
3 | 2 | csbeq2i 3901 | . 2 ⊢ ⦋𝐴 / 𝑥⦌-𝐵 = ⦋𝐴 / 𝑥⦌(0 − 𝐵) |
4 | df-neg 11444 | . 2 ⊢ -⦋𝐴 / 𝑥⦌𝐵 = (0 − ⦋𝐴 / 𝑥⦌𝐵) | |
5 | 1, 3, 4 | 3eqtr4g 2798 | 1 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌-𝐵 = -⦋𝐴 / 𝑥⦌𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 ⦋csb 3893 (class class class)co 7406 0cc0 11107 − cmin 11441 -cneg 11442 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-dm 5686 df-iota 6493 df-fv 6549 df-ov 7409 df-neg 11444 |
This theorem is referenced by: dvfsum2 25543 renegclALT 37822 |
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