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Theorem qseq2 7354
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 7341 . . . . 5  |-  ( A  =  B  ->  [ x ] A  =  [
x ] B )
21eqeq2d 2468 . . . 4  |-  ( A  =  B  ->  (
y  =  [ x ] A  <->  y  =  [
x ] B ) )
32rexbidv 2965 . . 3  |-  ( A  =  B  ->  ( E. x  e.  C  y  =  [ x ] A  <->  E. x  e.  C  y  =  [ x ] B ) )
43abbidv 2590 . 2  |-  ( A  =  B  ->  { y  |  E. x  e.  C  y  =  [
x ] A }  =  { y  |  E. x  e.  C  y  =  [ x ] B } )
5 df-qs 7309 . 2  |-  ( C /. A )  =  { y  |  E. x  e.  C  y  =  [ x ] A }
6 df-qs 7309 . 2  |-  ( C /. B )  =  { y  |  E. x  e.  C  y  =  [ x ] B }
74, 5, 63eqtr4g 2520 1  |-  ( A  =  B  ->  ( C /. A )  =  ( C /. B
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1398   {cab 2439   E.wrex 2805   [cec 7301   /.cqs 7302
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-br 4440  df-opab 4498  df-cnv 4996  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-ec 7305  df-qs 7309
This theorem is referenced by:  pi1bas3  21709  pstmval  28109
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