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Theorem ssiin 3952
Description: Subset theorem for an indexed intersection. (Contributed by NM, 15-Oct-2003.)
Assertion
Ref Expression
ssiin  |-  ( C 
C_  |^|_ x  e.  A  B 
<-> 
A. x  e.  A  C  C_  B )
Distinct variable group:    x, C
Allowed substitution hints:    A( x)    B( x)

Proof of Theorem ssiin
StepHypRef Expression
1 nfcv 2419 . 2  |-  F/_ x C
21ssiinf 3951 1  |-  ( C 
C_  |^|_ x  e.  A  B 
<-> 
A. x  e.  A  C  C_  B )
Colors of variables: wff set class
Syntax hints:    <-> wb 176   A.wral 2543    C_ wss 3152   |^|_ciin 3906
This theorem is referenced by:  cflim2  7889  ptbasfi  17276  limciun  19244  inttop2  25515  clsint2  26247  fnemeet2  26316  dihglblem4  31487  dihglblem6  31530
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-iin 3908
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