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Mirrors > Home > MPE Home > Th. List > opeq1d | Structured version Visualization version GIF version |
Description: Equality deduction for ordered pairs. (Contributed by NM, 16-Dec-2006.) |
Ref | Expression |
---|---|
opeq1d.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
Ref | Expression |
---|---|
opeq1d | ⊢ (𝜑 → 〈𝐴, 𝐶〉 = 〈𝐵, 𝐶〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opeq1d.1 | . 2 ⊢ (𝜑 → 𝐴 = 𝐵) | |
2 | opeq1 4340 | . 2 ⊢ (𝐴 = 𝐵 → 〈𝐴, 𝐶〉 = 〈𝐵, 𝐶〉) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝜑 → 〈𝐴, 𝐶〉 = 〈𝐵, 𝐶〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 〈cop 4131 |
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-3an 1033 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-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 |
This theorem is referenced by: oteq1 4349 oteq2 4350 opth 4871 elsnxp 5594 cbvoprab2 6626 unxpdomlem1 8049 mulcanenq 9661 ax1rid 9861 axrnegex 9862 fseq1m1p1 12284 uzrdglem 12618 swrd0swrd 13313 swrdccat 13344 swrdccat3a 13345 swrdccat3blem 13346 cshw0 13391 cshwmodn 13392 s2prop 13502 s4prop 13505 fsum2dlem 14343 fprod2dlem 14549 ruclem1 14799 imasaddvallem 16012 iscatd2 16165 moni 16219 homadmcd 16515 curf1 16688 curf1cl 16691 curf2 16692 hofcl 16722 gsum2dlem2 18193 imasdsf1olem 21988 ovoliunlem1 23077 cxpcn3 24289 axlowdimlem15 25636 axlowdim 25641 nvi 26853 nvop 26915 phop 27057 br8d 28802 fgreu 28854 1stpreimas 28866 smatfval 29189 smatrcl 29190 smatlem 29191 fvproj 29227 mvhfval 30684 mpst123 30691 br8 30899 fvtransport 31309 rfovcnvf1od 37318 |
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