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Mirrors > Home > ILE Home > Th. List > negeqd | GIF version |
Description: Equality deduction for negatives. (Contributed by NM, 14-May-1999.) |
Ref | Expression |
---|---|
negeqd.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
Ref | Expression |
---|---|
negeqd | ⊢ (𝜑 → -𝐴 = -𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | negeqd.1 | . 2 ⊢ (𝜑 → 𝐴 = 𝐵) | |
2 | negeq 8168 | . 2 ⊢ (𝐴 = 𝐵 → -𝐴 = -𝐵) | |
3 | 1, 2 | syl 14 | 1 ⊢ (𝜑 → -𝐴 = -𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1364 -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: negdi 8232 mulneg2 8371 mulm1 8375 eqord2 8459 mulreim 8579 apneg 8586 divnegap 8681 div2negap 8710 recgt0 8825 infrenegsupex 9612 supminfex 9615 mul2lt0rlt0 9777 ceilqval 10324 ceilid 10333 modqcyc2 10378 monoord2 10495 reneg 10895 imneg 10903 cjcj 10910 cjneg 10917 minmax 11256 minabs 11262 telfsumo2 11493 sinneg 11752 tannegap 11754 sincossq 11774 odd2np1 11896 oexpneg 11900 modgcd 12010 pcneg 12342 mulgval 13030 mulgneg 13046 ivthdec 14519 limcimolemlt 14530 dvrecap 14574 sinperlem 14626 efimpi 14637 ptolemy 14642 lgsneg1 14823 lgseisenlem1 14847 m1lgs 14849 ex-ceil 14875 |
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