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Theorem opthg 4712
 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 4202 . . . 4
21eqeq1d 2445 . . 3
3 eqeq1 2447 . . . 4
43anbi1d 704 . . 3
52, 4bibi12d 321 . 2
6 opeq2 4203 . . . 4
76eqeq1d 2445 . . 3
8 eqeq1 2447 . . . 4
98anbi2d 703 . . 3
107, 9bibi12d 321 . 2
11 vex 3098 . . 3
12 vex 3098 . . 3
1311, 12opth 4711 . 2
145, 10, 13vtocl2g 3157 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 184   wa 369   wceq 1383   wcel 1804  cop 4020 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pr 4676 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-rab 2802  df-v 3097  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021 This theorem is referenced by:  opth1g  4713  opthg2  4714  opthneg  4716  otthg  4720  oteqex  4730  s111  12604  symg2bas  16401  frgpnabllem1  16855  frgpnabllem2  16856  mat1dimbas  18951  el2wlkonotot0  24848  dvheveccl  36579
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