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Theorem ssiinf 3951
Description: Subset theorem for an indexed intersection. (Contributed by FL, 15-Oct-2012.) (Proof shortened by Mario Carneiro, 14-Oct-2016.)
Hypothesis
Ref Expression
ssiinf.1  |-  F/_ x C
Assertion
Ref Expression
ssiinf  |-  ( C 
C_  |^|_ x  e.  A  B 
<-> 
A. x  e.  A  C  C_  B )

Proof of Theorem ssiinf
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 vex 2791 . . . . 5  |-  y  e. 
_V
2 eliin 3910 . . . . 5  |-  ( y  e.  _V  ->  (
y  e.  |^|_ x  e.  A  B  <->  A. x  e.  A  y  e.  B ) )
31, 2ax-mp 8 . . . 4  |-  ( y  e.  |^|_ x  e.  A  B 
<-> 
A. x  e.  A  y  e.  B )
43ralbii 2567 . . 3  |-  ( A. y  e.  C  y  e.  |^|_ x  e.  A  B 
<-> 
A. y  e.  C  A. x  e.  A  y  e.  B )
5 ssiinf.1 . . . 4  |-  F/_ x C
6 nfcv 2419 . . . 4  |-  F/_ y A
75, 6ralcomf 2698 . . 3  |-  ( A. y  e.  C  A. x  e.  A  y  e.  B  <->  A. x  e.  A  A. y  e.  C  y  e.  B )
84, 7bitri 240 . 2  |-  ( A. y  e.  C  y  e.  |^|_ x  e.  A  B 
<-> 
A. x  e.  A  A. y  e.  C  y  e.  B )
9 dfss3 3170 . 2  |-  ( C 
C_  |^|_ x  e.  A  B 
<-> 
A. y  e.  C  y  e.  |^|_ x  e.  A  B )
10 dfss3 3170 . . 3  |-  ( C 
C_  B  <->  A. y  e.  C  y  e.  B )
1110ralbii 2567 . 2  |-  ( A. x  e.  A  C  C_  B  <->  A. x  e.  A  A. y  e.  C  y  e.  B )
128, 9, 113bitr4i 268 1  |-  ( C 
C_  |^|_ x  e.  A  B 
<-> 
A. x  e.  A  C  C_  B )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    e. wcel 1684   F/_wnfc 2406   A.wral 2543   _Vcvv 2788    C_ wss 3152   |^|_ciin 3906
This theorem is referenced by:  ssiin  3952  dmiin  4922
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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