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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 7204 | . 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 1243 -cneg 7183 |
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 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-rex 2312 df-v 2559 df-un 2922 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-br 3765 df-iota 4867 df-fv 4910 df-ov 5515 df-neg 7185 |
This theorem is referenced by: negdi 7268 mulneg2 7393 mulm1 7397 mulreim 7595 apneg 7602 divnegap 7683 div2negap 7711 recgt0 7816 ceilqval 9148 ceilid 9157 monoord2 9236 reneg 9468 imneg 9476 cjcj 9483 cjneg 9490 ex-ceil 9896 |
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