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Theorem ssiin 4141
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 2572 . 2  |-  F/_ x C
21ssiinf 4140 1  |-  ( C 
C_  |^|_ x  e.  A  B 
<-> 
A. x  e.  A  C  C_  B )
Colors of variables: wff set class
Syntax hints:    <-> wb 177   A.wral 2705    C_ wss 3320   |^|_ciin 4094
This theorem is referenced by:  cflim2  8143  ptbasfi  17613  limciun  19781  clsint2  26332  fnemeet2  26396  dihglblem4  32095  dihglblem6  32138
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 2417
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 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ral 2710  df-v 2958  df-in 3327  df-ss 3334  df-iin 4096
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