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Theorem ssiinf 4348
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 3083 . . . . 5  |-  y  e. 
_V
2 eliin 4305 . . . . 5  |-  ( y  e.  _V  ->  (
y  e.  |^|_ x  e.  A  B  <->  A. x  e.  A  y  e.  B ) )
31, 2ax-mp 5 . . . 4  |-  ( y  e.  |^|_ x  e.  A  B 
<-> 
A. x  e.  A  y  e.  B )
43ralbii 2853 . . 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 2580 . . . 4  |-  F/_ y A
75, 6ralcomf 2984 . . 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 252 . 2  |-  ( A. y  e.  C  y  e.  |^|_ x  e.  A  B 
<-> 
A. x  e.  A  A. y  e.  C  y  e.  B )
9 dfss3 3454 . 2  |-  ( C 
C_  |^|_ x  e.  A  B 
<-> 
A. y  e.  C  y  e.  |^|_ x  e.  A  B )
10 dfss3 3454 . . 3  |-  ( C 
C_  B  <->  A. y  e.  C  y  e.  B )
1110ralbii 2853 . 2  |-  ( A. x  e.  A  C  C_  B  <->  A. x  e.  A  A. y  e.  C  y  e.  B )
128, 9, 113bitr4i 280 1  |-  ( C 
C_  |^|_ x  e.  A  B 
<-> 
A. x  e.  A  C  C_  B )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 187    e. wcel 1872   F/_wnfc 2566   A.wral 2771   _Vcvv 3080    C_ wss 3436   |^|_ciin 4300
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ral 2776  df-v 3082  df-in 3443  df-ss 3450  df-iin 4302
This theorem is referenced by:  ssiin  4349  dmiin  5097
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