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Mirrors > Home > ILE Home > Th. List > negeq | GIF version |
Description: Equality theorem for negatives. (Contributed by NM, 10-Feb-1995.) |
Ref | Expression |
---|---|
negeq | ⊢ (A = B → -A = -B) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq2 5463 | . 2 ⊢ (A = B → (0 − A) = (0 − B)) | |
2 | df-neg 6982 | . 2 ⊢ -A = (0 − A) | |
3 | df-neg 6982 | . 2 ⊢ -B = (0 − B) | |
4 | 1, 2, 3 | 3eqtr4g 2094 | 1 ⊢ (A = B → -A = -B) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1242 (class class class)co 5455 0cc0 6711 − cmin 6979 -cneg 6980 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-rex 2306 df-v 2553 df-un 2916 df-sn 3373 df-pr 3374 df-op 3376 df-uni 3572 df-br 3756 df-iota 4810 df-fv 4853 df-ov 5458 df-neg 6982 |
This theorem is referenced by: negeqi 7002 negeqd 7003 neg11 7058 recexre 7362 elz 8023 znegcl 8052 zaddcllemneg 8060 elz2 8088 zindd 8132 ublbneg 8324 eqreznegel 8325 negm 8326 qnegcl 8347 xnegeq 8510 expival 8911 expnegap0 8917 m1expcl2 8931 |
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