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Theorem altopeq12 25799
Description: Equality for alternate ordered pairs. (Contributed by Scott Fenton, 22-Mar-2012.)
Assertion
Ref Expression
altopeq12  |-  ( ( A  =  B  /\  C  =  D )  -> 
<< A ,  C >>  = 
<< B ,  D >> )

Proof of Theorem altopeq12
StepHypRef Expression
1 sneq 3817 . . 3  |-  ( A  =  B  ->  { A }  =  { B } )
2 sneq 3817 . . 3  |-  ( C  =  D  ->  { C }  =  { D } )
31, 2anim12i 550 . 2  |-  ( ( A  =  B  /\  C  =  D )  ->  ( { A }  =  { B }  /\  { C }  =  { D } ) )
4 altopthsn 25798 . 2  |-  ( << A ,  C >>  =  << B ,  D >>  <->  ( { A }  =  { B }  /\  { C }  =  { D } ) )
53, 4sylibr 204 1  |-  ( ( A  =  B  /\  C  =  D )  -> 
<< A ,  C >>  = 
<< B ,  D >> )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652   {csn 3806   <<caltop 25793
This theorem is referenced by:  altopeq1  25800  altopeq2  25801
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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