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Theorem csbnegg 10470
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 6854 . 2 (𝐴𝑉𝐴 / 𝑥(0 − 𝐵) = (0 − 𝐴 / 𝑥𝐵))
2 df-neg 10461 . . 3 -𝐵 = (0 − 𝐵)
32csbeq2i 4136 . 2 𝐴 / 𝑥-𝐵 = 𝐴 / 𝑥(0 − 𝐵)
4 df-neg 10461 . 2 -𝐴 / 𝑥𝐵 = (0 − 𝐴 / 𝑥𝐵)
51, 3, 43eqtr4g 2819 1 (𝐴𝑉𝐴 / 𝑥-𝐵 = -𝐴 / 𝑥𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1632  wcel 2139  csb 3674  (class class class)co 6813  0cc0 10128  cmin 10458  -cneg 10459
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-nul 4941  ax-pow 4992
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-fal 1638  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-dm 5276  df-iota 6012  df-fv 6057  df-ov 6816  df-neg 10461
This theorem is referenced by:  dvfsum2  23996  renegclALT  34752
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