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Theorem qseq2 6529
Description: Equality theorem for quotient set. (Contributed by NM, 23-Jul-1995.)
Assertion
Ref Expression
qseq2  |-  ( A  =  B  ->  ( C /. A )  =  ( C /. B
) )

Proof of Theorem qseq2
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eceq2 6517 . . . . 5  |-  ( A  =  B  ->  [ x ] A  =  [
x ] B )
21eqeq2d 2169 . . . 4  |-  ( A  =  B  ->  (
y  =  [ x ] A  <->  y  =  [
x ] B ) )
32rexbidv 2458 . . 3  |-  ( A  =  B  ->  ( E. x  e.  C  y  =  [ x ] A  <->  E. x  e.  C  y  =  [ x ] B ) )
43abbidv 2275 . 2  |-  ( A  =  B  ->  { y  |  E. x  e.  C  y  =  [
x ] A }  =  { y  |  E. x  e.  C  y  =  [ x ] B } )
5 df-qs 6486 . 2  |-  ( C /. A )  =  { y  |  E. x  e.  C  y  =  [ x ] A }
6 df-qs 6486 . 2  |-  ( C /. B )  =  { y  |  E. x  e.  C  y  =  [ x ] B }
74, 5, 63eqtr4g 2215 1  |-  ( A  =  B  ->  ( C /. A )  =  ( C /. B
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1335   {cab 2143   E.wrex 2436   [cec 6478   /.cqs 6479
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-rex 2441  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-sn 3566  df-pr 3567  df-op 3569  df-br 3966  df-opab 4026  df-cnv 4594  df-dm 4596  df-rn 4597  df-res 4598  df-ima 4599  df-ec 6482  df-qs 6486
This theorem is referenced by: (None)
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