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Theorem opthg 3975
 Description: Ordered pair theorem. and are not required to be sets under our specific ordered pair definition. (Contributed by NM, 14-Oct-2005.) (Revised by Mario Carneiro, 26-Apr-2015.)
Assertion
Ref Expression
opthg

Proof of Theorem opthg
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 opeq1 3549 . . . 4
21eqeq1d 2048 . . 3
3 eqeq1 2046 . . . 4
43anbi1d 438 . . 3
52, 4bibi12d 224 . 2
6 opeq2 3550 . . . 4
76eqeq1d 2048 . . 3
8 eqeq1 2046 . . . 4
98anbi2d 437 . . 3
107, 9bibi12d 224 . 2
11 vex 2560 . . 3
12 vex 2560 . . 3
1311, 12opth 3974 . 2
145, 10, 13vtocl2g 2617 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 97   wb 98   wceq 1243   wcel 1393  cop 3378 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384 This theorem is referenced by:  opthg2  3976  xpopth  5802  eqop  5803  preqlu  6570  cauappcvgprlemladd  6756  elrealeu  6906
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