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Mirrors > Home > MPE Home > Th. List > ineq1i | Structured version Visualization version GIF version |
Description: Equality inference for intersection of two classes. (Contributed by NM, 26-Dec-1993.) |
Ref | Expression |
---|---|
ineq1i.1 | ⊢ 𝐴 = 𝐵 |
Ref | Expression |
---|---|
ineq1i | ⊢ (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ineq1i.1 | . 2 ⊢ 𝐴 = 𝐵 | |
2 | ineq1 4221 | . 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 1537 ∩ cin 3962 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-rab 3434 df-in 3970 |
This theorem is referenced by: in12 4237 inindi 4243 dfrab3 4325 dfif5 4547 disjpr2 4718 disjtpsn 4720 disjtp2 4721 resres 6013 imainrect 6203 predidm 6349 fresaun 6780 fresaunres2 6781 ssenen 9190 hartogslem1 9580 prinfzo0 13735 leiso 14495 f1oun2prg 14953 smumul 16527 setsfun 17205 setsfun0 17206 firest 17479 lsmdisj2r 19718 frgpuplem 19805 ltbwe 22080 tgrest 23183 fiuncmp 23428 ptclsg 23639 metnrmlem3 24897 mbfid 25684 ppi1 27222 cht1 27223 ppiub 27263 lrrecse 27990 lrrecpred 27992 chdmj2i 31511 chjassi 31515 pjoml2i 31614 pjoml4i 31616 cmcmlem 31620 mayetes3i 31758 cvmdi 32353 atomli 32411 atabsi 32430 uniin1 32572 disjuniel 32617 imadifxp 32621 gtiso 32716 preiman0 32725 nn0disj01 32825 prsss 33877 ordtrest2NEW 33884 esumnul 34029 measinblem 34201 eulerpartlemt 34353 ballotlem2 34470 ballotlemfp1 34473 ballotlemfval0 34477 chtvalz 34623 fmla0disjsuc 35383 mthmpps 35567 dffv5 35906 bj-sscon 37012 bj-discrmoore 37094 mblfinlem2 37645 ismblfin 37648 mbfposadd 37654 itg2addnclem2 37659 asindmre 37690 abeqin 38234 xrnres 38384 redundeq1 38611 refrelsredund4 38614 diophrw 42747 dnwech 43037 lmhmlnmsplit 43076 rp-fakeuninass 43506 iunrelexp0 43692 nznngen 44312 uzinico2 45515 limsup0 45650 limsupvaluz 45664 sge0sn 46335 31prm 47522 |
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