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Theorem ssintab 3705
Description: Subclass of the intersection of a class abstraction. (Contributed by NM, 31-Jul-2006.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
Assertion
Ref Expression
ssintab  |-  ( A 
C_  |^| { x  | 
ph }  <->  A. x
( ph  ->  A  C_  x ) )
Distinct variable group:    x, A
Allowed substitution hint:    ph( x)

Proof of Theorem ssintab
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 ssint 3704 . 2  |-  ( A 
C_  |^| { x  | 
ph }  <->  A. y  e.  { x  |  ph } A  C_  y )
2 sseq2 3048 . . 3  |-  ( y  =  x  ->  ( A  C_  y  <->  A  C_  x
) )
32ralab2 2779 . 2  |-  ( A. y  e.  { x  |  ph } A  C_  y 
<-> 
A. x ( ph  ->  A  C_  x )
)
41, 3bitri 182 1  |-  ( A 
C_  |^| { x  | 
ph }  <->  A. x
( ph  ->  A  C_  x ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103   A.wal 1287   {cab 2074   A.wral 2359    C_ wss 2999   |^|cint 3688
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-v 2621  df-in 3005  df-ss 3012  df-int 3689
This theorem is referenced by:  ssmin  3707  ssintrab  3711  intmin4  3716  dfuzi  8854
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