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Theorem csbnegg 10870
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 7186 . 2 (𝐴𝑉𝐴 / 𝑥(0 − 𝐵) = (0 − 𝐴 / 𝑥𝐵))
2 df-neg 10860 . . 3 -𝐵 = (0 − 𝐵)
32csbeq2i 3873 . 2 𝐴 / 𝑥-𝐵 = 𝐴 / 𝑥(0 − 𝐵)
4 df-neg 10860 . 2 -𝐴 / 𝑥𝐵 = (0 − 𝐴 / 𝑥𝐵)
51, 3, 43eqtr4g 2884 1 (𝐴𝑉𝐴 / 𝑥-𝐵 = -𝐴 / 𝑥𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2115  csb 3865  (class class class)co 7140  0cc0 10524  cmin 10857  -cneg 10858
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-nul 5193  ax-pow 5249
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3137  df-rex 3138  df-rab 3141  df-v 3481  df-sbc 3758  df-csb 3866  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-nul 4275  df-if 4449  df-sn 4549  df-pr 4551  df-op 4555  df-uni 4822  df-br 5050  df-dm 5548  df-iota 6297  df-fv 6346  df-ov 7143  df-neg 10860
This theorem is referenced by:  dvfsum2  24628  renegclALT  36164
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