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Theorem opthg 4167
 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 3712 . . . 4
21eqeq1d 2149 . . 3
3 eqeq1 2147 . . . 4
43anbi1d 461 . . 3
52, 4bibi12d 234 . 2
6 opeq2 3713 . . . 4
76eqeq1d 2149 . . 3
8 eqeq1 2147 . . . 4
98anbi2d 460 . . 3
107, 9bibi12d 234 . 2
11 vex 2692 . . 3
12 vex 2692 . . 3
1311, 12opth 4166 . 2
145, 10, 13vtocl2g 2753 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 103   wb 104   wceq 1332   wcel 1481  cop 3534 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4053  ax-pow 4105  ax-pr 4138 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-un 3079  df-in 3081  df-ss 3088  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540 This theorem is referenced by:  opthg2  4168  xpopth  6081  eqop  6082  inl11  6957  preqlu  7303  cauappcvgprlemladd  7489  elrealeu  7660  qnumdenbi  11904  crth  11934
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