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Theorem pweq 3513
Description: Equality theorem for power class. (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
pweq  |-  ( A  =  B  ->  ~P A  =  ~P B
)

Proof of Theorem pweq
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 sseq2 3121 . . 3  |-  ( A  =  B  ->  (
x  C_  A  <->  x  C_  B
) )
21abbidv 2257 . 2  |-  ( A  =  B  ->  { x  |  x  C_  A }  =  { x  |  x 
C_  B } )
3 df-pw 3512 . 2  |-  ~P A  =  { x  |  x 
C_  A }
4 df-pw 3512 . 2  |-  ~P B  =  { x  |  x 
C_  B }
52, 3, 43eqtr4g 2197 1  |-  ( A  =  B  ->  ~P A  =  ~P B
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1331   {cab 2125    C_ wss 3071   ~Pcpw 3510
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-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-in 3077  df-ss 3084  df-pw 3512
This theorem is referenced by:  pweqi  3514  pweqd  3515  axpweq  4095  pwexg  4104  pwssunim  4206  ordpwsucexmid  4485  fival  6858  istopg  12176  istopon  12190  eltg  12231  tgdom  12251  ntrval  12289
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