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Theorem qseq1 7373
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 3064 . . 3  |-  ( A  =  B  ->  ( E. x  e.  A  y  =  [ x ] C  <->  E. x  e.  B  y  =  [ x ] C ) )
21abbidv 2603 . 2  |-  ( A  =  B  ->  { y  |  E. x  e.  A  y  =  [
x ] C }  =  { y  |  E. x  e.  B  y  =  [ x ] C } )
3 df-qs 7329 . 2  |-  ( A /. C )  =  { y  |  E. x  e.  A  y  =  [ x ] C }
4 df-qs 7329 . 2  |-  ( B /. C )  =  { y  |  E. x  e.  B  y  =  [ x ] C }
52, 3, 43eqtr4g 2533 1  |-  ( A  =  B  ->  ( A /. C )  =  ( B /. C
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379   {cab 2452   E.wrex 2818   [cec 7321   /.cqs 7322
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-rex 2823  df-qs 7329
This theorem is referenced by:  pi1bas  21404  pstmval  27706
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