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Theorem altopthc 25523
Description: Alternate ordered pair theorem with different sethood requirements. See altopth 25521 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 2382 . 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 25522 . 2  |-  ( << C ,  D >>  =  << A ,  B >>  <->  ( C  =  A  /\  D  =  B ) )
5 eqcom 2382 . . 3  |-  ( C  =  A  <->  A  =  C )
6 eqcom 2382 . . 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 1717   _Vcvv 2892   <<caltop 25508
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 2361  ax-sep 4264  ax-nul 4272  ax-pr 4337
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 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-v 2894  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-sn 3756  df-pr 3757  df-altop 25510
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