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Theorem ssintab 4027
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 4026 . 2  |-  ( A 
C_  |^| { x  | 
ph }  <->  A. y  e.  { x  |  ph } A  C_  y )
2 sseq2 3330 . . 3  |-  ( y  =  x  ->  ( A  C_  y  <->  A  C_  x
) )
32ralab2 3059 . 2  |-  ( A. y  e.  { x  |  ph } A  C_  y 
<-> 
A. x ( ph  ->  A  C_  x )
)
41, 3bitri 241 1  |-  ( A 
C_  |^| { x  | 
ph }  <->  A. x
( ph  ->  A  C_  x ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177   A.wal 1546   {cab 2390   A.wral 2666    C_ wss 3280   |^|cint 4010
This theorem is referenced by:  ssmin  4029  ssintrab  4033  intmin4  4039  dffi2  7386  rankval3b  7708  sstskm  8673  dfuzi  10316  cycsubg  14923
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ral 2671  df-v 2918  df-in 3287  df-ss 3294  df-int 4011
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