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Theorem altopthb 28137
Description: Alternate ordered pair theorem with different sethood requirements. See altopth 28136 for more comments. (Contributed by Scott Fenton, 14-Apr-2012.)
Hypotheses
Ref Expression
altopthb.1  |-  A  e. 
_V
altopthb.2  |-  D  e. 
_V
Assertion
Ref Expression
altopthb  |-  ( << A ,  B >>  =  << C ,  D >>  <->  ( A  =  C  /\  B  =  D ) )

Proof of Theorem altopthb
StepHypRef Expression
1 altopthb.1 . 2  |-  A  e. 
_V
2 altopthb.2 . 2  |-  D  e. 
_V
3 altopthbg 28135 . 2  |-  ( ( A  e.  _V  /\  D  e.  _V )  ->  ( << A ,  B >>  =  << C ,  D >>  <->  ( A  =  C  /\  B  =  D )
) )
41, 2, 3mp2an 672 1  |-  ( << A ,  B >>  =  << C ,  D >>  <->  ( A  =  C  /\  B  =  D ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758   _Vcvv 3070   <<caltop 28123
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4513  ax-nul 4521  ax-pr 4631
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-v 3072  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-sn 3978  df-pr 3980  df-altop 28125
This theorem is referenced by:  altopthc  28138
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