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Theorem pweq 3428
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 3046 . . 3  |-  ( A  =  B  ->  (
x  C_  A  <->  x  C_  B
) )
21abbidv 2205 . 2  |-  ( A  =  B  ->  { x  |  x  C_  A }  =  { x  |  x 
C_  B } )
3 df-pw 3427 . 2  |-  ~P A  =  { x  |  x 
C_  A }
4 df-pw 3427 . 2  |-  ~P B  =  { x  |  x 
C_  B }
52, 3, 43eqtr4g 2145 1  |-  ( A  =  B  ->  ~P A  =  ~P B
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1289   {cab 2074    C_ wss 2997   ~Pcpw 3425
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-11 1442  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-in 3003  df-ss 3010  df-pw 3427
This theorem is referenced by:  pweqi  3429  pweqd  3430  axpweq  3998  pwexg  4007  pwssunim  4102  ordpwsucexmid  4376
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