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Mirrors > Home > MPE Home > Th. List > negeqi | Structured version Visualization version GIF version |
Description: Equality inference for negatives. (Contributed by NM, 14-Feb-1995.) |
Ref | Expression |
---|---|
negeqi.1 | ⊢ 𝐴 = 𝐵 |
Ref | Expression |
---|---|
negeqi | ⊢ -𝐴 = -𝐵 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | negeqi.1 | . 2 ⊢ 𝐴 = 𝐵 | |
2 | negeq 11429 | . 2 ⊢ (𝐴 = 𝐵 → -𝐴 = -𝐵) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ -𝐴 = -𝐵 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1541 -cneg 11422 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2702 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2709 df-cleq 2723 df-clel 2809 df-rab 3429 df-v 3471 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-nul 4314 df-if 4518 df-sn 4618 df-pr 4620 df-op 4624 df-uni 4897 df-br 5137 df-iota 6479 df-fv 6535 df-ov 7391 df-neg 11424 |
This theorem is referenced by: negsubdii 11522 recgt0ii 12097 m1expcl2 14028 crreczi 14168 absi 15210 geo2sum2 15797 bpoly2 15978 bpoly3 15979 sinhval 16074 coshval 16075 cos2bnd 16108 divalglem2 16315 m1expaddsub 19325 cnmsgnsubg 21056 psgninv 21061 ncvspi 24595 cphipval2 24680 ditg0 25292 cbvditg 25293 ang180lem2 26235 ang180lem3 26236 ang180lem4 26237 1cubrlem 26266 dcubic2 26269 atandm2 26302 efiasin 26313 asinsinlem 26316 asinsin 26317 asin1 26319 reasinsin 26321 atancj 26335 atantayl2 26363 ppiub 26627 lgseisenlem1 26798 lgseisenlem2 26799 lgsquadlem1 26803 ostth3 27061 nvpi 29729 ipidsq 29772 ipasslem10 29901 normlem1 30172 polid2i 30219 lnophmlem2 31079 archirngz 32152 xrge0iif1 32684 ballotlem2 33253 itg2addnclem3 36280 dvasin 36311 areacirc 36320 lhe4.4ex1a 42796 itgsin0pilem1 44376 stoweidlem26 44452 dirkertrigeqlem3 44526 fourierdlem103 44635 sqwvfourb 44655 fourierswlem 44656 proththd 45991 |
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