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Theorem opthg 4246
 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 3796 . . . 4
21eqeq1d 2291 . . 3
3 eqeq1 2289 . . . 4
43anbi1d 685 . . 3
52, 4bibi12d 312 . 2
6 opeq2 3797 . . . 4
76eqeq1d 2291 . . 3
8 eqeq1 2289 . . . 4
98anbi2d 684 . . 3
107, 9bibi12d 312 . 2
11 vex 2791 . . 3
12 vex 2791 . . 3
1311, 12opth 4245 . 2
145, 10, 13vtocl2g 2847 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 176   wa 358   wceq 1623   wcel 1684  cop 3643 This theorem is referenced by:  opthg2  4247  oteqex  4259  s111  11448  frgpnabllem1  15161  frgpnabllem2  15162  dvheveccl  31302 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649
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