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Theorem opthg2 4380
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 4379 . 2  |-  ( ( C  e.  V  /\  D  e.  W )  ->  ( <. C ,  D >.  =  <. A ,  B >.  <-> 
( C  =  A  /\  D  =  B ) ) )
2 eqcom 2391 . 2  |-  ( <. A ,  B >.  = 
<. C ,  D >.  <->  <. C ,  D >.  =  <. A ,  B >. )
3 eqcom 2391 . . 3  |-  ( A  =  C  <->  C  =  A )
4 eqcom 2391 . . 3  |-  ( B  =  D  <->  D  =  B )
53, 4anbi12i 679 . 2  |-  ( ( A  =  C  /\  B  =  D )  <->  ( C  =  A  /\  D  =  B )
)
61, 2, 53bitr4g 280 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    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717   <.cop 3762
This theorem is referenced by:  opth2  4381  fliftel  5972
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pr 4346
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-rab 2660  df-v 2903  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-sn 3765  df-pr 3766  df-op 3768
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