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Theorem altopth2 25803
Description: Equality of the second members of equal alternate ordered pairs, which holds regardless of the first members' sethood. (Contributed by Scott Fenton, 22-Mar-2012.)
Assertion
Ref Expression
altopth2  |-  ( B  e.  V  ->  ( << A ,  B >>  = 
<< C ,  D >>  ->  B  =  D )
)

Proof of Theorem altopth2
StepHypRef Expression
1 altopthsn 25798 . 2  |-  ( << A ,  B >>  =  << C ,  D >>  <->  ( { A }  =  { C }  /\  { B }  =  { D } ) )
2 sneqrg 3960 . . 3  |-  ( B  e.  V  ->  ( { B }  =  { D }  ->  B  =  D ) )
32adantld 454 . 2  |-  ( B  e.  V  ->  (
( { A }  =  { C }  /\  { B }  =  { D } )  ->  B  =  D ) )
41, 3syl5bi 209 1  |-  ( B  e.  V  ->  ( << A ,  B >>  = 
<< C ,  D >>  ->  B  =  D )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   {csn 3806   <<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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