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Theorem opthg 4727
 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 4217 . . . 4
21eqeq1d 2459 . . 3
3 eqeq1 2461 . . . 4
43anbi1d 704 . . 3
52, 4bibi12d 321 . 2
6 opeq2 4218 . . . 4
76eqeq1d 2459 . . 3
8 eqeq1 2461 . . . 4
98anbi2d 703 . . 3
107, 9bibi12d 321 . 2
11 vex 3112 . . 3
12 vex 3112 . . 3
1311, 12opth 4726 . 2
145, 10, 13vtocl2g 3171 1
 Colors of variables: wff setvar class Syntax hints:  ->wi 4  <->wb 184  /\wa 369  =wceq 1395  e.wcel 1818  <.cop 4035 This theorem is referenced by:  opth1g  4728  opthg2  4729  opthneg  4731  otthg  4735  oteqex  4745  s111  12623  symg2bas  16423  frgpnabllem1  16877  frgpnabllem2  16878  mat1dimbas  18974  el2wlkonotot0  24872  embedsetcestrclem  32663  dvheveccl  36839 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pr 4691 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-rab 2816  df-v 3111  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036
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