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Mirrors > Home > ILE Home > Th. List > opeq2 | Unicode version |
Description: Equality theorem for ordered pairs. (Contributed by NM, 25-Jun-1998.) (Revised by Mario Carneiro, 26-Apr-2015.) |
Ref | Expression |
---|---|
opeq2 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq1 2240 |
. . . . . 6
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2 | 1 | anbi2d 464 |
. . . . 5
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3 | eqidd 2178 |
. . . . . . 7
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4 | preq2 3670 |
. . . . . . 7
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5 | 3, 4 | preq12d 3677 |
. . . . . 6
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6 | 5 | eleq2d 2247 |
. . . . 5
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7 | 2, 6 | anbi12d 473 |
. . . 4
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8 | df-3an 980 |
. . . 4
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9 | df-3an 980 |
. . . 4
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10 | 7, 8, 9 | 3bitr4g 223 |
. . 3
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11 | 10 | abbidv 2295 |
. 2
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12 | df-op 3601 |
. 2
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13 | df-op 3601 |
. 2
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14 | 11, 12, 13 | 3eqtr4g 2235 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-v 2739 df-un 3133 df-sn 3598 df-pr 3599 df-op 3601 |
This theorem is referenced by: opeq12 3780 opeq2i 3782 opeq2d 3785 oteq2 3788 oteq3 3789 breq2 4007 cbvopab2 4077 cbvopab2v 4080 opthg 4238 eqvinop 4243 opelopabsb 4260 opelxp 4656 opabid2 4758 elrn2g 4817 opeldm 4830 opeldmg 4832 elrn2 4869 opelresg 4914 iss 4953 elimasng 4996 issref 5011 dmsnopg 5100 cnvsng 5114 elxp4 5116 elxp5 5117 dffun5r 5228 funopg 5250 f1osng 5502 tz6.12f 5544 fsn 5688 fsng 5689 fvsng 5712 oveq2 5882 cbvoprab2 5947 ovg 6012 opabex3d 6121 opabex3 6122 op1stg 6150 op2ndg 6151 oprssdmm 6171 op1steq 6179 dfoprab4f 6193 tfrlemibxssdm 6327 tfr1onlembxssdm 6343 tfrcllembxssdm 6356 elixpsn 6734 ixpsnf1o 6735 mapsnen 6810 xpsnen 6820 xpassen 6829 xpf1o 6843 djulclr 7047 djurclr 7048 djulcl 7049 djurcl 7050 djulclb 7053 inl11 7063 djuss 7068 1stinl 7072 2ndinl 7073 1stinr 7074 2ndinr 7075 elreal 7826 ax1rid 7875 fseq1p1m1 10091 imasaddfnlemg 12717 cnmpt21 13684 djucllem 14434 |
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