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Theorem altopth 25529
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 4377), 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 25527 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( << A ,  B >>  =  << C ,  D >>  <->  ( A  =  C  /\  B  =  D )
) )
41, 2, 3mp2an 654 1  |-  ( << A ,  B >>  =  << C ,  D >>  <->  ( A  =  C  /\  B  =  D ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717   _Vcvv 2900   <<caltop 25516
This theorem is referenced by:  altopthd  25532  altopelaltxp  25536
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 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pr 4345
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 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-v 2902  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-sn 3764  df-pr 3765  df-altop 25518
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