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Theorem ssintab 3632
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 3631 . 2  |-  ( A 
C_  |^| { x  | 
ph }  <->  A. y  e.  { x  |  ph } A  C_  y )
2 sseq2 2967 . . 3  |-  ( y  =  x  ->  ( A  C_  y  <->  A  C_  x
) )
32ralab2 2705 . 2  |-  ( A. y  e.  { x  |  ph } A  C_  y 
<-> 
A. x ( ph  ->  A  C_  x )
)
41, 3bitri 173 1  |-  ( A 
C_  |^| { x  | 
ph }  <->  A. x
( ph  ->  A  C_  x ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 98   A.wal 1241   {cab 2026   A.wral 2306    C_ wss 2917   |^|cint 3615
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-v 2559  df-in 2924  df-ss 2931  df-int 3616
This theorem is referenced by:  ssmin  3634  ssintrab  3638  intmin4  3643  dfuzi  8348
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