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Theorem qseq2 6955
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 6942 . . . . 5  |-  ( A  =  B  ->  [ x ] A  =  [
x ] B )
21eqeq2d 2447 . . . 4  |-  ( A  =  B  ->  (
y  =  [ x ] A  <->  y  =  [
x ] B ) )
32rexbidv 2726 . . 3  |-  ( A  =  B  ->  ( E. x  e.  C  y  =  [ x ] A  <->  E. x  e.  C  y  =  [ x ] B ) )
43abbidv 2550 . 2  |-  ( A  =  B  ->  { y  |  E. x  e.  C  y  =  [
x ] A }  =  { y  |  E. x  e.  C  y  =  [ x ] B } )
5 df-qs 6911 . 2  |-  ( C /. A )  =  { y  |  E. x  e.  C  y  =  [ x ] A }
6 df-qs 6911 . 2  |-  ( C /. B )  =  { y  |  E. x  e.  C  y  =  [ x ] B }
74, 5, 63eqtr4g 2493 1  |-  ( A  =  B  ->  ( C /. A )  =  ( C /. B
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652   {cab 2422   E.wrex 2706   [cec 6903   /.cqs 6904
This theorem is referenced by:  pi1bas3  19068  pstmval  24290
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-br 4213  df-opab 4267  df-cnv 4886  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-ec 6907  df-qs 6911
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