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Theorem altopthbg 25061
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 25054 . 2  |-  ( << A ,  B >>  =  << C ,  D >>  <->  ( { A }  =  { C }  /\  { B }  =  { D } ) )
2 sneqbg 3864 . . 3  |-  ( A  e.  V  ->  ( { A }  =  { C }  <->  A  =  C
) )
3 sneqbg 3864 . . . 4  |-  ( D  e.  W  ->  ( { D }  =  { B }  <->  D  =  B
) )
4 eqcom 2360 . . . 4  |-  ( { B }  =  { D }  <->  { D }  =  { B } )
5 eqcom 2360 . . . 4  |-  ( B  =  D  <->  D  =  B )
63, 4, 53bitr4g 279 . . 3  |-  ( D  e.  W  ->  ( { B }  =  { D }  <->  B  =  D
) )
72, 6bi2anan9 843 . 2  |-  ( ( A  e.  V  /\  D  e.  W )  ->  ( ( { A }  =  { C }  /\  { B }  =  { D } )  <-> 
( A  =  C  /\  B  =  D ) ) )
81, 7syl5bb 248 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 176    /\ wa 358    = wceq 1642    e. wcel 1710   {csn 3716   <<caltop 25049
This theorem is referenced by:  altopthb  25063
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pr 4295
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-v 2866  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-sn 3722  df-pr 3723  df-altop 25051
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