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| Description: If the difference between two numbers is zero, they are equal. (Contributed by NM, 16-Nov-1999.) | 
| Ref | Expression | 
|---|---|
| subeq0 | ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 − 𝐵) = 0 ↔ 𝐴 = 𝐵)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | subid 11529 | . . . 4 ⊢ (𝐵 ∈ ℂ → (𝐵 − 𝐵) = 0) | |
| 2 | 1 | adantl 481 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐵 − 𝐵) = 0) | 
| 3 | 2 | eqeq2d 2747 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 − 𝐵) = (𝐵 − 𝐵) ↔ (𝐴 − 𝐵) = 0)) | 
| 4 | subcan2 11535 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 − 𝐵) = (𝐵 − 𝐵) ↔ 𝐴 = 𝐵)) | |
| 5 | 4 | 3anidm23 1422 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 − 𝐵) = (𝐵 − 𝐵) ↔ 𝐴 = 𝐵)) | 
| 6 | 3, 5 | bitr3d 281 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 − 𝐵) = 0 ↔ 𝐴 = 𝐵)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1539 ∈ wcel 2107 (class class class)co 7432 ℂcc 11154 0cc0 11156 − cmin 11493 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-po 5591 df-so 5592 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-er 8746 df-en 8987 df-dom 8988 df-sdom 8989 df-pnf 11298 df-mnf 11299 df-ltxr 11301 df-sub 11495 | 
| This theorem is referenced by: subeq0i 11590 subeq0d 11629 subne0d 11630 subeq0ad 11631 mulcan1g 11917 div2sub 12093 cju 12263 nn0sub 12578 addmodlteq 13988 geoserg 15903 geolim 15907 geolim2 15908 georeclim 15909 geoisum1c 15917 tanadd 16204 fzocongeq 16362 divalglem8 16438 mndodcongi 19562 odf1 19581 odf1o1 19591 cnmet 24793 iccpnfhmeo 24977 plyremlem 26347 geolim3 26382 abelthlem2 26477 abelthlem7 26483 efeq1 26571 tanregt0 26582 logtayl 26703 ang180lem1 26853 ang180lem2 26854 ang180lem3 26855 lawcos 26860 isosctrlem1 26862 isosctrlem2 26863 atandm2 26921 atandm4 26923 2efiatan 26962 tanatan 26963 dvatan 26979 mumullem2 27224 mersenne 27272 dchrsum2 27313 sumdchr2 27315 addsq2reu 27485 axcgrid 28932 axcontlem2 28981 hvmulcan2 31093 poimirlem13 37641 rencldnfilem 42836 qirropth 42924 dvconstbi 44358 isosctrlem1ALT 44959 rrx2pnedifcoorneor 48642 rrx2pnedifcoorneorr 48643 | 
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