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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 11431 | . 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 11424 |
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 3430 df-v 3472 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4520 df-sn 4620 df-pr 4622 df-op 4626 df-uni 4899 df-br 5139 df-iota 6481 df-fv 6537 df-ov 7393 df-neg 11426 |
This theorem is referenced by: negsubdii 11524 recgt0ii 12099 m1expcl2 14030 crreczi 14170 absi 15212 geo2sum2 15799 bpoly2 15980 bpoly3 15981 sinhval 16076 coshval 16077 cos2bnd 16110 divalglem2 16317 m1expaddsub 19327 cnmsgnsubg 21058 psgninv 21063 ncvspi 24597 cphipval2 24682 ditg0 25294 cbvditg 25295 ang180lem2 26237 ang180lem3 26238 ang180lem4 26239 1cubrlem 26268 dcubic2 26271 atandm2 26304 efiasin 26315 asinsinlem 26318 asinsin 26319 asin1 26321 reasinsin 26323 atancj 26337 atantayl2 26365 ppiub 26629 lgseisenlem1 26800 lgseisenlem2 26801 lgsquadlem1 26805 ostth3 27063 nvpi 29778 ipidsq 29821 ipasslem10 29950 normlem1 30221 polid2i 30268 lnophmlem2 31128 archirngz 32201 xrge0iif1 32733 ballotlem2 33302 itg2addnclem3 36329 dvasin 36360 areacirc 36369 lhe4.4ex1a 42845 itgsin0pilem1 44425 stoweidlem26 44501 dirkertrigeqlem3 44575 fourierdlem103 44684 sqwvfourb 44704 fourierswlem 44705 proththd 46040 |
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