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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 5899 | . 2 ⊢ (𝐴 = 𝐵 → (0 − 𝐴) = (0 − 𝐵)) | |
2 | df-neg 8149 | . 2 ⊢ -𝐴 = (0 − 𝐴) | |
3 | df-neg 8149 | . 2 ⊢ -𝐵 = (0 − 𝐵) | |
4 | 1, 2, 3 | 3eqtr4g 2247 | 1 ⊢ (𝐴 = 𝐵 → -𝐴 = -𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1364 (class class class)co 5891 0cc0 7829 − cmin 8146 -cneg 8147 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-rex 2474 df-v 2754 df-un 3148 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-br 4019 df-iota 5193 df-fv 5239 df-ov 5894 df-neg 8149 |
This theorem is referenced by: negeqi 8169 negeqd 8170 neg11 8226 negf1o 8357 recexre 8553 negiso 8930 elz 9273 znegcl 9302 zaddcllemneg 9310 elz2 9342 zindd 9389 infrenegsupex 9612 supinfneg 9613 infsupneg 9614 supminfex 9615 ublbneg 9631 eqreznegel 9632 negm 9633 qnegcl 9654 xnegeq 9845 ceilqval 10324 exp3val 10540 expnegap0 10546 m1expcl2 10560 negfi 11254 dvdsnegb 11833 infssuzex 11968 infssuzcldc 11970 zsupssdc 11973 lcmneg 12092 pcexp 12327 pcneg 12342 znnen 12417 mulgneg2 13062 negcncf 14485 negfcncf 14486 lgsdir2lem4 14829 ex-ceil 14875 |
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