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Theorem altopthg 24573
Description: Alternate ordered pair theorem. (Contributed by Scott Fenton, 22-Mar-2012.)
Assertion
Ref Expression
altopthg  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( << A ,  B >>  =  << C ,  D >>  <->  ( A  =  C  /\  B  =  D )
) )

Proof of Theorem altopthg
StepHypRef Expression
1 altopthsn 24567 . 2  |-  ( << A ,  B >>  =  << C ,  D >>  <->  ( { A }  =  { C }  /\  { B }  =  { D } ) )
2 sneqbg 3799 . . 3  |-  ( A  e.  V  ->  ( { A }  =  { C }  <->  A  =  C
) )
3 sneqbg 3799 . . 3  |-  ( B  e.  W  ->  ( { B }  =  { D }  <->  B  =  D
) )
42, 3bi2anan9 843 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ( { A }  =  { C }  /\  { B }  =  { D } )  <-> 
( A  =  C  /\  B  =  D ) ) )
51, 4syl5bb 248 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( << A ,  B >>  =  << C ,  D >>  <->  ( A  =  C  /\  B  =  D )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   {csn 3653   <<caltop 24562
This theorem is referenced by:  altopth  24575
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-sn 3659  df-pr 3660  df-altop 24564
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