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Theorem csbnegg 10157
 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.)
Assertion
Ref Expression
csbnegg (𝐴𝑉𝐴 / 𝑥-𝐵 = -𝐴 / 𝑥𝐵)

Proof of Theorem csbnegg
StepHypRef Expression
1 csbov2g 6589 . 2 (𝐴𝑉𝐴 / 𝑥(0 − 𝐵) = (0 − 𝐴 / 𝑥𝐵))
2 df-neg 10148 . . 3 -𝐵 = (0 − 𝐵)
32csbeq2i 3945 . 2 𝐴 / 𝑥-𝐵 = 𝐴 / 𝑥(0 − 𝐵)
4 df-neg 10148 . 2 -𝐴 / 𝑥𝐵 = (0 − 𝐴 / 𝑥𝐵)
51, 3, 43eqtr4g 2669 1 (𝐴𝑉𝐴 / 𝑥-𝐵 = -𝐴 / 𝑥𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  ⦋csb 3499  (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-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-nul 4717  ax-pow 4769 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  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-sbc 3403  df-csb 3500  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-dm 5048  df-iota 5768  df-fv 5812  df-ov 6552  df-neg 10148 This theorem is referenced by:  dvfsum2  23601  renegclALT  33267
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