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Theorem pwssb 4169
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 4168 . 2  |-  ( A 
C_  ~P B  <->  U. A  C_  B )
2 unissb 4037 . 2  |-  ( U. A  C_  B  <->  A. x  e.  A  x  C_  B
)
31, 2bitri 241 1  |-  ( A 
C_  ~P B  <->  A. x  e.  A  x  C_  B
)
Colors of variables: wff set class
Syntax hints:    <-> wb 177   A.wral 2697    C_ wss 3312   ~Pcpw 3791   U.cuni 4007
This theorem is referenced by:  istps5OLD  16981  ustuni  18248  metustfbasOLD  18587  metustfbas  18588  dmvlsiga  24504  1stmbfm  24602  2ndmbfm  24603  dya2iocucvr  24626
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702  df-v 2950  df-in 3319  df-ss 3326  df-pw 3793  df-uni 4008
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