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Theorem altopth 24503
Description: The alternate ordered pair theorem. If two alternate ordered pairs are equal, their first elements are equal and their second elements are equal. Note that  C and  D are not required to be a set due to a peculiarity of our specific ordered pair definition, as opposed to the regular ordered pairs used here, which (as in opth 4245), requires  D to be a set. (Contributed by Scott Fenton, 23-Mar-2012.)
Hypotheses
Ref Expression
altopth.1  |-  A  e. 
_V
altopth.2  |-  B  e. 
_V
Assertion
Ref Expression
altopth  |-  ( << A ,  B >>  =  << C ,  D >>  <->  ( A  =  C  /\  B  =  D ) )

Proof of Theorem altopth
StepHypRef Expression
1 altopth.1 . 2  |-  A  e. 
_V
2 altopth.2 . 2  |-  B  e. 
_V
3 altopthg 24501 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( << A ,  B >>  =  << C ,  D >>  <->  ( A  =  C  /\  B  =  D )
) )
41, 2, 3mp2an 653 1  |-  ( << A ,  B >>  =  << C ,  D >>  <->  ( A  =  C  /\  B  =  D ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788   <<caltop 24490
This theorem is referenced by:  altopthd  24506  altopelaltxp  24510
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-sn 3646  df-pr 3647  df-altop 24492
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