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Theorem qseq2 7172
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 7159 . . . . 5  |-  ( A  =  B  ->  [ x ] A  =  [
x ] B )
21eqeq2d 2454 . . . 4  |-  ( A  =  B  ->  (
y  =  [ x ] A  <->  y  =  [
x ] B ) )
32rexbidv 2757 . . 3  |-  ( A  =  B  ->  ( E. x  e.  C  y  =  [ x ] A  <->  E. x  e.  C  y  =  [ x ] B ) )
43abbidv 2563 . 2  |-  ( A  =  B  ->  { y  |  E. x  e.  C  y  =  [
x ] A }  =  { y  |  E. x  e.  C  y  =  [ x ] B } )
5 df-qs 7128 . 2  |-  ( C /. A )  =  { y  |  E. x  e.  C  y  =  [ x ] A }
6 df-qs 7128 . 2  |-  ( C /. B )  =  { y  |  E. x  e.  C  y  =  [ x ] B }
74, 5, 63eqtr4g 2500 1  |-  ( A  =  B  ->  ( C /. A )  =  ( C /. B
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1369   {cab 2429   E.wrex 2737   [cec 7120   /.cqs 7121
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-rex 2742  df-rab 2745  df-v 2995  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-nul 3659  df-if 3813  df-sn 3899  df-pr 3901  df-op 3905  df-br 4314  df-opab 4372  df-cnv 4869  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-ec 7124  df-qs 7128
This theorem is referenced by:  pi1bas3  20637  pstmval  26344
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