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Theorem qseq1 6286
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 2559 . . 3  |-  ( A  =  B  ->  ( E. x  e.  A  y  =  [ x ] C  <->  E. x  e.  B  y  =  [ x ] C ) )
21abbidv 2202 . 2  |-  ( A  =  B  ->  { y  |  E. x  e.  A  y  =  [
x ] C }  =  { y  |  E. x  e.  B  y  =  [ x ] C } )
3 df-qs 6244 . 2  |-  ( A /. C )  =  { y  |  E. x  e.  A  y  =  [ x ] C }
4 df-qs 6244 . 2  |-  ( B /. C )  =  { y  |  E. x  e.  B  y  =  [ x ] C }
52, 3, 43eqtr4g 2142 1  |-  ( A  =  B  ->  ( A /. C )  =  ( B /. C
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1287   {cab 2071   E.wrex 2356   [cec 6236   /.cqs 6237
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-rex 2361  df-qs 6244
This theorem is referenced by: (None)
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