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Theorem altopthd 25819
Description: Alternate ordered pair theorem with different sethood requirements. See altopth 25816 for more comments. (Contributed by Scott Fenton, 14-Apr-2012.)
Hypotheses
Ref Expression
altopthd.1  |-  C  e. 
_V
altopthd.2  |-  D  e. 
_V
Assertion
Ref Expression
altopthd  |-  ( << A ,  B >>  =  << C ,  D >>  <->  ( A  =  C  /\  B  =  D ) )

Proof of Theorem altopthd
StepHypRef Expression
1 eqcom 2440 . 2  |-  ( << A ,  B >>  =  << C ,  D >>  <->  << C ,  D >>  =  << A ,  B >> )
2 altopthd.1 . . 3  |-  C  e. 
_V
3 altopthd.2 . . 3  |-  D  e. 
_V
42, 3altopth 25816 . 2  |-  ( << C ,  D >>  =  << A ,  B >>  <->  ( C  =  A  /\  D  =  B ) )
5 eqcom 2440 . . 3  |-  ( C  =  A  <->  A  =  C )
6 eqcom 2440 . . 3  |-  ( D  =  B  <->  B  =  D )
75, 6anbi12i 680 . 2  |-  ( ( C  =  A  /\  D  =  B )  <->  ( A  =  C  /\  B  =  D )
)
81, 4, 73bitri 264 1  |-  ( << A ,  B >>  =  << C ,  D >>  <->  ( A  =  C  /\  B  =  D ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   _Vcvv 2958   <<caltop 25803
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-sn 3822  df-pr 3823  df-altop 25805
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