Theorem csbnegg 7306

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	⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌-𝐵 = -⦋𝐴 / 𝑥⦌𝐵)

Proof of Theorem csbnegg

Step	Hyp	Ref	Expression
1		csbov2g 5566	. 2 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(0 − 𝐵) = (0 − ⦋𝐴 / 𝑥⦌𝐵))
2		df-neg 7282	. . 3 ⊢ -𝐵 = (0 − 𝐵)
3	2	csbeq2i 2932	. 2 ⊢ ⦋𝐴 / 𝑥⦌-𝐵 = ⦋𝐴 / 𝑥⦌(0 − 𝐵)
4		df-neg 7282	. 2 ⊢ -⦋𝐴 / 𝑥⦌𝐵 = (0 − ⦋𝐴 / 𝑥⦌𝐵)
5	1, 3, 4	3eqtr4g 2138	1 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌-𝐵 = -⦋𝐴 / 𝑥⦌𝐵)

Syntax hints:   → wi 4   = wceq 1284   ∈ wcel 1433  ⦋csb 2908  (class class class)co 5532  0cc0 6981   − cmin 7279  -cneg 7280

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 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-rex 2354  df-v 2603  df-sbc 2816  df-csb 2909  df-un 2977  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-iota 4887  df-fv 4930  df-ov 5535  df-neg 7282

This theorem is referenced by: (None)