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Mirrors > Home > MPE Home > Th. List > ineq12d | Structured version Visualization version GIF version |
Description: Equality deduction for intersection of two classes. (Contributed by NM, 24-Jun-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) |
Ref | Expression |
---|---|
ineq1d.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
ineq12d.2 | ⊢ (𝜑 → 𝐶 = 𝐷) |
Ref | Expression |
---|---|
ineq12d | ⊢ (𝜑 → (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ineq1d.1 | . 2 ⊢ (𝜑 → 𝐴 = 𝐵) | |
2 | ineq12d.2 | . 2 ⊢ (𝜑 → 𝐶 = 𝐷) | |
3 | ineq12 3771 | . 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐷)) | |
4 | 1, 2, 3 | syl2anc 691 | 1 ⊢ (𝜑 → (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∩ cin 3539 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-v 3175 df-in 3547 |
This theorem is referenced by: csbin 3962 predeq123 5598 funprgOLD 5855 funtpgOLD 5857 funcnvtp 5865 offval 6802 ofrfval 6803 oev2 7490 isf32lem7 9064 ressval 15754 invffval 16241 invfval 16242 dfiso2 16255 isofn 16258 oppcinv 16263 zerooval 16472 isps 17025 dmdprd 18220 dprddisj 18231 dprdf1o 18254 dmdprdsplit2lem 18267 dmdprdpr 18271 pgpfaclem1 18303 isunit 18480 dfrhm2 18540 isrhm 18544 2idlval 19054 aspval 19149 ressmplbas2 19276 pjfval 19869 iscon 21026 consuba 21033 ptbasin 21190 ptclsg 21228 qtopval 21308 rnelfmlem 21566 trust 21843 isnmhm 22360 uniioombllem2a 23156 dyaddisjlem 23169 dyaddisj 23170 i1faddlem 23266 i1fmullem 23267 limcflf 23451 ispth 26098 1pthonlem2 26120 2pthlem2 26126 constr2pth 26131 constr3pthlem3 26185 frisusgranb 26524 2spotdisj 26588 chocin 27738 cmbr3 27851 pjoml3 27855 fh1 27861 xppreima2 28830 hauseqcn 29269 prsssdm 29291 ordtrestNEW 29295 ordtrest2NEW 29297 cndprobval 29822 ballotlemfrc 29915 bnj1421 30364 msrval 30689 msrf 30693 ismfs 30700 clsun 31493 poimirlem8 32587 itg2addnclem2 32632 heiborlem4 32783 heiborlem6 32785 heiborlem10 32789 pmodl42N 34155 polfvalN 34208 poldmj1N 34232 pmapj2N 34233 pnonsingN 34237 psubclinN 34252 poml4N 34257 osumcllem9N 34268 trnfsetN 34460 diainN 35364 djaffvalN 35440 djafvalN 35441 djajN 35444 dihmeetcl 35652 dihmeet2 35653 dochnoncon 35698 djhffval 35703 djhfval 35704 djhlj 35708 dochdmm1 35717 lclkrlem2g 35820 lclkrlem2v 35835 lcfrlem21 35870 lcfrlem24 35873 mapdunirnN 35957 baerlem5amN 36023 baerlem5bmN 36024 baerlem5abmN 36025 mapdheq4lem 36038 mapdh6lem1N 36040 mapdh6lem2N 36041 hdmap1l6lem1 36115 hdmap1l6lem2 36116 aomclem8 36649 csbingOLD 38076 disjrnmpt2 38370 dvsinax 38801 dvcosax 38816 nnfoctbdjlem 39348 ewlksfval 40803 isewlk 40804 ewlkinedg 40806 isPth 40929 trlsegvdeg 41395 frcond3 41440 rhmval 41709 |
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