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Theorem qseq1 6956
Description: Equality theorem for quotient set. (Contributed by NM, 23-Jul-1995.)
Assertion
Ref Expression
qseq1  |-  ( A  =  B  ->  ( A /. C )  =  ( B /. C
) )

Proof of Theorem qseq1
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rexeq 2907 . . 3  |-  ( A  =  B  ->  ( E. x  e.  A  y  =  [ x ] C  <->  E. x  e.  B  y  =  [ x ] C ) )
21abbidv 2552 . 2  |-  ( A  =  B  ->  { y  |  E. x  e.  A  y  =  [
x ] C }  =  { y  |  E. x  e.  B  y  =  [ x ] C } )
3 df-qs 6913 . 2  |-  ( A /. C )  =  { y  |  E. x  e.  A  y  =  [ x ] C }
4 df-qs 6913 . 2  |-  ( B /. C )  =  { y  |  E. x  e.  B  y  =  [ x ] C }
52, 3, 43eqtr4g 2495 1  |-  ( A  =  B  ->  ( A /. C )  =  ( B /. C
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653   {cab 2424   E.wrex 2708   [cec 6905   /.cqs 6906
This theorem is referenced by:  pi1bas  19065  pstmval  24292
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-rex 2713  df-qs 6913
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