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Theorem altopthbg 28004
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 27997 . 2  |-  ( << A ,  B >>  =  << C ,  D >>  <->  ( { A }  =  { C }  /\  { B }  =  { D } ) )
2 sneqbg 4048 . . 3  |-  ( A  e.  V  ->  ( { A }  =  { C }  <->  A  =  C
) )
3 sneqbg 4048 . . . 4  |-  ( D  e.  W  ->  ( { D }  =  { B }  <->  D  =  B
) )
4 eqcom 2445 . . . 4  |-  ( { B }  =  { D }  <->  { D }  =  { B } )
5 eqcom 2445 . . . 4  |-  ( B  =  D  <->  D  =  B )
63, 4, 53bitr4g 288 . . 3  |-  ( D  e.  W  ->  ( { B }  =  { D }  <->  B  =  D
) )
72, 6bi2anan9 868 . 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 1369    e. wcel 1756   {csn 3882   <<caltop 27992
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pr 4536
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-v 2979  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-sn 3883  df-pr 3885  df-altop 27994
This theorem is referenced by:  altopthb  28006
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