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Theorem altopthc 25808
Description: Alternate ordered pair theorem with different sethood requirements. See altopth 25806 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 2437 . 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 25807 . 2  |-  ( << C ,  D >>  =  << A ,  B >>  <->  ( C  =  A  /\  D  =  B ) )
5 eqcom 2437 . . 3  |-  ( C  =  A  <->  A  =  C )
6 eqcom 2437 . . 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 1652    e. wcel 1725   _Vcvv 2948   <<caltop 25793
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-sn 3812  df-pr 3813  df-altop 25795
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