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Theorem pwssb 3988
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 3987 . 2  |-  ( A 
C_  ~P B  <->  U. A  C_  B )
2 unissb 3857 . 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 2543    C_ wss 3152   ~Pcpw 3625   U.cuni 3827
This theorem is referenced by:  istps5OLD  16662  dmvlsiga  23490  1stmbfm  23565  2ndmbfm  23566
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-v 2790  df-in 3159  df-ss 3166  df-pw 3627  df-uni 3828
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