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Theorem altopeq12 25522
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 3769 . . 3  |-  ( A  =  B  ->  { A }  =  { B } )
2 sneq 3769 . . 3  |-  ( C  =  D  ->  { C }  =  { D } )
31, 2anim12i 550 . 2  |-  ( ( A  =  B  /\  C  =  D )  ->  ( { A }  =  { B }  /\  { C }  =  { D } ) )
4 altopthsn 25521 . 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 1649   {csn 3758   <<caltop 25516
This theorem is referenced by:  altopeq1  25523  altopeq2  25524
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 2369  ax-sep 4272  ax-nul 4280  ax-pr 4345
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 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-v 2902  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-sn 3764  df-pr 3765  df-altop 25518
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