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Theorem opthg2 4066
Description: Ordered pair theorem. (Contributed by NM, 14-Oct-2005.) (Revised by Mario Carneiro, 26-Apr-2015.)
Assertion
Ref Expression
opthg2  |-  ( ( C  e.  V  /\  D  e.  W )  ->  ( <. A ,  B >.  =  <. C ,  D >.  <-> 
( A  =  C  /\  B  =  D ) ) )

Proof of Theorem opthg2
StepHypRef Expression
1 opthg 4065 . 2  |-  ( ( C  e.  V  /\  D  e.  W )  ->  ( <. C ,  D >.  =  <. A ,  B >.  <-> 
( C  =  A  /\  D  =  B ) ) )
2 eqcom 2090 . 2  |-  ( <. A ,  B >.  = 
<. C ,  D >.  <->  <. C ,  D >.  =  <. A ,  B >. )
3 eqcom 2090 . . 3  |-  ( A  =  C  <->  C  =  A )
4 eqcom 2090 . . 3  |-  ( B  =  D  <->  D  =  B )
53, 4anbi12i 448 . 2  |-  ( ( A  =  C  /\  B  =  D )  <->  ( C  =  A  /\  D  =  B )
)
61, 2, 53bitr4g 221 1  |-  ( ( C  e.  V  /\  D  e.  W )  ->  ( <. A ,  B >.  =  <. C ,  D >.  <-> 
( A  =  C  /\  B  =  D ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1289    e. wcel 1438   <.cop 3449
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455
This theorem is referenced by:  opth2  4067  fliftel  5572  axprecex  7415
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