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Theorem ssiin 4329
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 2616 . 2  |-  F/_ x C
21ssiinf 4328 1  |-  ( C 
C_  |^|_ x  e.  A  B 
<-> 
A. x  e.  A  C  C_  B )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184   A.wral 2799    C_ wss 3437   |^|_ciin 4281
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ral 2804  df-v 3080  df-in 3444  df-ss 3451  df-iin 4283
This theorem is referenced by:  cflim2  8544  ptbasfi  19287  limciun  21503  clsint2  28673  fnemeet2  28737  dihglblem4  35281  dihglblem6  35324
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