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Theorem altopthbg 25725
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 25718 . 2  |-  ( << A ,  B >>  =  << C ,  D >>  <->  ( { A }  =  { C }  /\  { B }  =  { D } ) )
2 sneqbg 3937 . . 3  |-  ( A  e.  V  ->  ( { A }  =  { C }  <->  A  =  C
) )
3 sneqbg 3937 . . . 4  |-  ( D  e.  W  ->  ( { D }  =  { B }  <->  D  =  B
) )
4 eqcom 2414 . . . 4  |-  ( { B }  =  { D }  <->  { D }  =  { B } )
5 eqcom 2414 . . . 4  |-  ( B  =  D  <->  D  =  B )
63, 4, 53bitr4g 280 . . 3  |-  ( D  e.  W  ->  ( { B }  =  { D }  <->  B  =  D
) )
72, 6bi2anan9 844 . 2  |-  ( ( A  e.  V  /\  D  e.  W )  ->  ( ( { A }  =  { C }  /\  { B }  =  { D } )  <-> 
( A  =  C  /\  B  =  D ) ) )
81, 7syl5bb 249 1  |-  ( ( A  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 1721   {csn 3782   <<caltop 25713
This theorem is referenced by:  altopthb  25727
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 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pr 4371
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-v 2926  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-sn 3788  df-pr 3789  df-altop 25715
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