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Theorem ssiin 4365
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 4364 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 2804    C_ wss 3461   |^|_ciin 4316
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ral 2809  df-v 3108  df-in 3468  df-ss 3475  df-iin 4318
This theorem is referenced by:  cflim2  8634  ptbasfi  20248  limciun  22464  clsint2  30387  fnemeet2  30425  dihglblem4  37421  dihglblem6  37464
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