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Theorem pwssb 4004
Description: Two ways to express a collection of subclasses. (Contributed by NM, 19-Jul-2006.)
Assertion
Ref Expression
pwssb  |-  ( A 
C_  ~P B  <->  A. x  e.  A  x  C_  B
)
Distinct variable groups:    x, A    x, B

Proof of Theorem pwssb
StepHypRef Expression
1 sspwuni 4003 . 2  |-  ( A 
C_  ~P B  <->  U. A  C_  B )
2 unissb 3873 . 2  |-  ( U. A  C_  B  <->  A. x  e.  A  x  C_  B
)
31, 2bitri 240 1  |-  ( A 
C_  ~P B  <->  A. x  e.  A  x  C_  B
)
Colors of variables: wff set class
Syntax hints:    <-> wb 176   A.wral 2556    C_ wss 3165   ~Pcpw 3638   U.cuni 3843
This theorem is referenced by:  istps5OLD  16678  dmvlsiga  23505  1stmbfm  23580  2ndmbfm  23581
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-v 2803  df-in 3172  df-ss 3179  df-pw 3640  df-uni 3844
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