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Theorem opthg 4246
Description: Ordered pair theorem.  C and  D 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  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( <. A ,  B >.  =  <. C ,  D >.  <-> 
( A  =  C  /\  B  =  D ) ) )

Proof of Theorem opthg
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 opeq1 3796 . . . 4  |-  ( x  =  A  ->  <. x ,  y >.  =  <. A ,  y >. )
21eqeq1d 2291 . . 3  |-  ( x  =  A  ->  ( <. x ,  y >.  =  <. C ,  D >.  <->  <. A ,  y >.  =  <. C ,  D >. ) )
3 eqeq1 2289 . . . 4  |-  ( x  =  A  ->  (
x  =  C  <->  A  =  C ) )
43anbi1d 685 . . 3  |-  ( x  =  A  ->  (
( x  =  C  /\  y  =  D )  <->  ( A  =  C  /\  y  =  D ) ) )
52, 4bibi12d 312 . 2  |-  ( x  =  A  ->  (
( <. x ,  y
>.  =  <. C ,  D >. 
<->  ( x  =  C  /\  y  =  D ) )  <->  ( <. A ,  y >.  =  <. C ,  D >.  <->  ( A  =  C  /\  y  =  D ) ) ) )
6 opeq2 3797 . . . 4  |-  ( y  =  B  ->  <. A , 
y >.  =  <. A ,  B >. )
76eqeq1d 2291 . . 3  |-  ( y  =  B  ->  ( <. A ,  y >.  =  <. C ,  D >.  <->  <. A ,  B >.  = 
<. C ,  D >. ) )
8 eqeq1 2289 . . . 4  |-  ( y  =  B  ->  (
y  =  D  <->  B  =  D ) )
98anbi2d 684 . . 3  |-  ( y  =  B  ->  (
( A  =  C  /\  y  =  D )  <->  ( A  =  C  /\  B  =  D ) ) )
107, 9bibi12d 312 . 2  |-  ( y  =  B  ->  (
( <. A ,  y
>.  =  <. C ,  D >. 
<->  ( A  =  C  /\  y  =  D ) )  <->  ( <. A ,  B >.  =  <. C ,  D >.  <->  ( A  =  C  /\  B  =  D ) ) ) )
11 vex 2791 . . 3  |-  x  e. 
_V
12 vex 2791 . . 3  |-  y  e. 
_V
1311, 12opth 4245 . 2  |-  ( <.
x ,  y >.  =  <. C ,  D >.  <-> 
( x  =  C  /\  y  =  D ) )
145, 10, 13vtocl2g 2847 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( <. A ,  B >.  =  <. C ,  D >.  <-> 
( A  =  C  /\  B  =  D ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. 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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