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Mirrors > Home > ILE Home > Th. List > csbnegg | GIF version |
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.) |
Ref | Expression |
---|---|
csbnegg | ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌-𝐵 = -⦋𝐴 / 𝑥⦌𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | csbov2g 5812 | . 2 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(0 − 𝐵) = (0 − ⦋𝐴 / 𝑥⦌𝐵)) | |
2 | df-neg 7936 | . . 3 ⊢ -𝐵 = (0 − 𝐵) | |
3 | 2 | csbeq2i 3029 | . 2 ⊢ ⦋𝐴 / 𝑥⦌-𝐵 = ⦋𝐴 / 𝑥⦌(0 − 𝐵) |
4 | df-neg 7936 | . 2 ⊢ -⦋𝐴 / 𝑥⦌𝐵 = (0 − ⦋𝐴 / 𝑥⦌𝐵) | |
5 | 1, 3, 4 | 3eqtr4g 2197 | 1 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌-𝐵 = -⦋𝐴 / 𝑥⦌𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1331 ∈ wcel 1480 ⦋csb 3003 (class class class)co 5774 0cc0 7620 − cmin 7933 -cneg 7934 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-rex 2422 df-v 2688 df-sbc 2910 df-csb 3004 df-un 3075 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-iota 5088 df-fv 5131 df-ov 5777 df-neg 7936 |
This theorem is referenced by: (None) |
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