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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 5782 | . 2 ⊢ (𝐴 = 𝐵 → (0 − 𝐴) = (0 − 𝐵)) | |
2 | df-neg 7936 | . 2 ⊢ -𝐴 = (0 − 𝐴) | |
3 | df-neg 7936 | . 2 ⊢ -𝐵 = (0 − 𝐵) | |
4 | 1, 2, 3 | 3eqtr4g 2197 | 1 ⊢ (𝐴 = 𝐵 → -𝐴 = -𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1331 (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-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: negeqi 7956 negeqd 7957 neg11 8013 negf1o 8144 recexre 8340 negiso 8713 elz 9056 znegcl 9085 zaddcllemneg 9093 elz2 9122 zindd 9169 infrenegsupex 9389 supinfneg 9390 infsupneg 9391 supminfex 9392 ublbneg 9405 eqreznegel 9406 negm 9407 qnegcl 9428 xnegeq 9610 ceilqval 10079 exp3val 10295 expnegap0 10301 m1expcl2 10315 negfi 10999 dvdsnegb 11510 infssuzex 11642 infssuzcldc 11644 lcmneg 11755 znnen 11911 negcncf 12757 negfcncf 12758 ex-ceil 12938 |
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