ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  qseq2 Unicode version

Theorem qseq2 6444
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 6432 . . . . 5  |-  ( A  =  B  ->  [ x ] A  =  [
x ] B )
21eqeq2d 2127 . . . 4  |-  ( A  =  B  ->  (
y  =  [ x ] A  <->  y  =  [
x ] B ) )
32rexbidv 2413 . . 3  |-  ( A  =  B  ->  ( E. x  e.  C  y  =  [ x ] A  <->  E. x  e.  C  y  =  [ x ] B ) )
43abbidv 2233 . 2  |-  ( A  =  B  ->  { y  |  E. x  e.  C  y  =  [
x ] A }  =  { y  |  E. x  e.  C  y  =  [ x ] B } )
5 df-qs 6401 . 2  |-  ( C /. A )  =  { y  |  E. x  e.  C  y  =  [ x ] A }
6 df-qs 6401 . 2  |-  ( C /. B )  =  { y  |  E. x  e.  C  y  =  [ x ] B }
74, 5, 63eqtr4g 2173 1  |-  ( A  =  B  ->  ( C /. A )  =  ( C /. B
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1314   {cab 2101   E.wrex 2392   [cec 6393   /.cqs 6394
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-rex 2397  df-v 2660  df-un 3043  df-in 3045  df-ss 3052  df-sn 3501  df-pr 3502  df-op 3504  df-br 3898  df-opab 3958  df-cnv 4515  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-ec 6397  df-qs 6401
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator