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Theorem ssintab 3879
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 3878 . 2  |-  ( A 
C_  |^| { x  | 
ph }  <->  A. y  e.  { x  |  ph } A  C_  y )
2 sseq2 3200 . . 3  |-  ( y  =  x  ->  ( A  C_  y  <->  A  C_  x
) )
32ralab2 2930 . 2  |-  ( A. y  e.  { x  |  ph } A  C_  y 
<-> 
A. x ( ph  ->  A  C_  x )
)
41, 3bitri 240 1  |-  ( A 
C_  |^| { x  | 
ph }  <->  A. x
( ph  ->  A  C_  x ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176   A.wal 1527   {cab 2269   A.wral 2543    C_ wss 3152   |^|cint 3862
This theorem is referenced by:  ssmin  3881  ssintrab  3885  intmin4  3891  dffi2  7176  rankval3b  7498  sstskm  8464  dfuzi  10102  cycsubg  14645  eqint  24960  eqintg  24961  xindcls  25997
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-int 3863
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