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Theorem pweq 3593
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 3194 . . 3  |-  ( A  =  B  ->  (
x  C_  A  <->  x  C_  B
) )
21abbidv 2307 . 2  |-  ( A  =  B  ->  { x  |  x  C_  A }  =  { x  |  x 
C_  B } )
3 df-pw 3592 . 2  |-  ~P A  =  { x  |  x 
C_  A }
4 df-pw 3592 . 2  |-  ~P B  =  { x  |  x 
C_  B }
52, 3, 43eqtr4g 2247 1  |-  ( A  =  B  ->  ~P A  =  ~P B
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1364   {cab 2175    C_ wss 3144   ~Pcpw 3590
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-in 3150  df-ss 3157  df-pw 3592
This theorem is referenced by:  pweqi  3594  pweqd  3595  axpweq  4189  pwexg  4198  pwssunim  4302  ordpwsucexmid  4587  exmidpw2en  6939  fival  6998  istopg  13951  istopon  13965  eltg  14004  tgdom  14024  ntrval  14062
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