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Theorem altopthc 25720
Description: Alternate ordered pair theorem with different sethood requirements. See altopth 25718 for more comments. (Contributed by Scott Fenton, 14-Apr-2012.)
Hypotheses
Ref Expression
altopthc.1  |-  B  e. 
_V
altopthc.2  |-  C  e. 
_V
Assertion
Ref Expression
altopthc  |-  ( << A ,  B >>  =  << C ,  D >>  <->  ( A  =  C  /\  B  =  D ) )

Proof of Theorem altopthc
StepHypRef Expression
1 eqcom 2406 . 2  |-  ( << A ,  B >>  =  << C ,  D >>  <->  << C ,  D >>  =  << A ,  B >> )
2 altopthc.2 . . 3  |-  C  e. 
_V
3 altopthc.1 . . 3  |-  B  e. 
_V
42, 3altopthb 25719 . 2  |-  ( << C ,  D >>  =  << A ,  B >>  <->  ( C  =  A  /\  D  =  B ) )
5 eqcom 2406 . . 3  |-  ( C  =  A  <->  A  =  C )
6 eqcom 2406 . . 3  |-  ( D  =  B  <->  B  =  D )
75, 6anbi12i 679 . 2  |-  ( ( C  =  A  /\  D  =  B )  <->  ( A  =  C  /\  B  =  D )
)
81, 4, 73bitri 263 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 1721   _Vcvv 2916   <<caltop 25705
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 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363
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 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-sn 3780  df-pr 3781  df-altop 25707
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