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Theorem csbnegg 11505
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 7479 . 2 (𝐴𝑉𝐴 / 𝑥(0 − 𝐵) = (0 − 𝐴 / 𝑥𝐵))
2 df-neg 11495 . . 3 -𝐵 = (0 − 𝐵)
32csbeq2i 3907 . 2 𝐴 / 𝑥-𝐵 = 𝐴 / 𝑥(0 − 𝐵)
4 df-neg 11495 . 2 -𝐴 / 𝑥𝐵 = (0 − 𝐴 / 𝑥𝐵)
51, 3, 43eqtr4g 2802 1 (𝐴𝑉𝐴 / 𝑥-𝐵 = -𝐴 / 𝑥𝐵)
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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