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Mirrors > Home > MPE Home > Th. List > difeq1i | Structured version Visualization version GIF version |
Description: Inference adding difference to the right in a class equality. (Contributed by NM, 15-Nov-2002.) |
Ref | Expression |
---|---|
difeq1i.1 | ⊢ 𝐴 = 𝐵 |
Ref | Expression |
---|---|
difeq1i | ⊢ (𝐴 ∖ 𝐶) = (𝐵 ∖ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | difeq1i.1 | . 2 ⊢ 𝐴 = 𝐵 | |
2 | difeq1 4128 | . 2 ⊢ (𝐴 = 𝐵 → (𝐴 ∖ 𝐶) = (𝐵 ∖ 𝐶)) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ (𝐴 ∖ 𝐶) = (𝐵 ∖ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1536 ∖ cdif 3959 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 |
This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-rab 3433 df-dif 3965 |
This theorem is referenced by: difeq12i 4133 dfin3 4282 indif1 4287 indifcom 4288 difun1 4304 notab 4319 rabdif 4326 notrab 4327 undifabs 4483 difprsn1 4804 difprsn2 4805 diftpsn3 4806 resdifcom 6018 resdmdfsn 6050 frpoind 6364 wfiOLD 6373 orddif 6481 fresaun 6779 f12dfv 7292 f13dfv 7293 domunsncan 9110 phplem1OLD 9251 elfiun 9467 frind 9787 dju1dif 10210 axcclem 10494 dfn2 12536 mvdco 19477 pmtrdifellem2 19509 islinds2 21850 lindsind2 21856 restcld 23195 ufprim 23932 volun 25593 itgsplitioo 25887 uhgr0vb 29103 uhgr0 29104 uvtxupgrres 29439 cplgr3v 29466 ex-dif 30451 indifundif 32551 imadifxp 32620 aciunf1 32679 pmtrcnelor 33093 lindsunlem 33651 lindsun 33652 braew 34222 carsgclctunlem1 34298 carsggect 34299 coinflippvt 34465 ballotlemfval0 34476 signstfvcl 34566 satf0 35356 onint1 36431 bj-2upln1upl 37006 bj-disj2r 37010 lindsenlbs 37601 poimirlem13 37619 poimirlem14 37620 poimirlem18 37624 poimirlem21 37627 poimirlem30 37636 itg2addnclem 37657 asindmre 37689 sucdifsn 38219 disjresundif 38223 ressucdifsn 38225 kelac2 43053 fourierdlem102 46163 fourierdlem114 46175 pwsal 46270 issald 46288 sge0fodjrnlem 46371 hoiprodp1 46543 lincext2 48300 disjdifb 48657 |
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