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Theorem csbnegg 7627
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 5649 . 2 (𝐴𝑉𝐴 / 𝑥(0 − 𝐵) = (0 − 𝐴 / 𝑥𝐵))
2 df-neg 7603 . . 3 -𝐵 = (0 − 𝐵)
32csbeq2i 2946 . 2 𝐴 / 𝑥-𝐵 = 𝐴 / 𝑥(0 − 𝐵)
4 df-neg 7603 . 2 -𝐴 / 𝑥𝐵 = (0 − 𝐴 / 𝑥𝐵)
51, 3, 43eqtr4g 2142 1 (𝐴𝑉𝐴 / 𝑥-𝐵 = -𝐴 / 𝑥𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1287  wcel 1436  csb 2922  (class class class)co 5615  0cc0 7297  cmin 7600  -cneg 7601
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-rex 2361  df-v 2617  df-sbc 2830  df-csb 2923  df-un 2992  df-sn 3437  df-pr 3438  df-op 3440  df-uni 3639  df-br 3823  df-iota 4948  df-fv 4991  df-ov 5618  df-neg 7603
This theorem is referenced by: (None)
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