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Theorem altopthbg 29545
Description: Alternate ordered pair theorem. (Contributed by Scott Fenton, 14-Apr-2012.)
Assertion
Ref Expression
altopthbg  |-  ( ( A  e.  V  /\  D  e.  W )  ->  ( << A ,  B >>  =  << C ,  D >>  <->  ( A  =  C  /\  B  =  D )
) )

Proof of Theorem altopthbg
StepHypRef Expression
1 altopthsn 29538 . 2  |-  ( << A ,  B >>  =  << C ,  D >>  <->  ( { A }  =  { C }  /\  { B }  =  { D } ) )
2 sneqbg 4203 . . 3  |-  ( A  e.  V  ->  ( { A }  =  { C }  <->  A  =  C
) )
3 sneqbg 4203 . . . 4  |-  ( D  e.  W  ->  ( { D }  =  { B }  <->  D  =  B
) )
4 eqcom 2476 . . . 4  |-  ( { B }  =  { D }  <->  { D }  =  { B } )
5 eqcom 2476 . . . 4  |-  ( B  =  D  <->  D  =  B )
63, 4, 53bitr4g 288 . . 3  |-  ( D  e.  W  ->  ( { B }  =  { D }  <->  B  =  D
) )
72, 6bi2anan9 871 . 2  |-  ( ( A  e.  V  /\  D  e.  W )  ->  ( ( { A }  =  { C }  /\  { B }  =  { D } )  <-> 
( A  =  C  /\  B  =  D ) ) )
81, 7syl5bb 257 1  |-  ( ( A  e.  V  /\  D  e.  W )  ->  ( << A ,  B >>  =  << C ,  D >>  <->  ( A  =  C  /\  B  =  D )
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   {csn 4033   <<caltop 29533
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pr 4692
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-v 3120  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-sn 4034  df-pr 4036  df-altop 29535
This theorem is referenced by:  altopthb  29547
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