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Theorem ssintrab 3901
Description: Subclass of the intersection of a restricted class builder. (Contributed by NM, 30-Jan-2015.)
Assertion
Ref Expression
ssintrab  |-  ( A 
C_  |^| { x  e.  B  |  ph }  <->  A. x  e.  B  (
ph  ->  A  C_  x
) )
Distinct variable group:    x, A
Allowed substitution hints:    ph( x)    B( x)

Proof of Theorem ssintrab
StepHypRef Expression
1 df-rab 2565 . . . 4  |-  { x  e.  B  |  ph }  =  { x  |  ( x  e.  B  /\  ph ) }
21inteqi 3882 . . 3  |-  |^| { x  e.  B  |  ph }  =  |^| { x  |  ( x  e.  B  /\  ph ) }
32sseq2i 3216 . 2  |-  ( A 
C_  |^| { x  e.  B  |  ph }  <->  A 
C_  |^| { x  |  ( x  e.  B  /\  ph ) } )
4 impexp 433 . . . 4  |-  ( ( ( x  e.  B  /\  ph )  ->  A  C_  x )  <->  ( x  e.  B  ->  ( ph  ->  A  C_  x )
) )
54albii 1556 . . 3  |-  ( A. x ( ( x  e.  B  /\  ph )  ->  A  C_  x
)  <->  A. x ( x  e.  B  ->  ( ph  ->  A  C_  x
) ) )
6 ssintab 3895 . . 3  |-  ( A 
C_  |^| { x  |  ( x  e.  B  /\  ph ) }  <->  A. x
( ( x  e.  B  /\  ph )  ->  A  C_  x )
)
7 df-ral 2561 . . 3  |-  ( A. x  e.  B  ( ph  ->  A  C_  x
)  <->  A. x ( x  e.  B  ->  ( ph  ->  A  C_  x
) ) )
85, 6, 73bitr4i 268 . 2  |-  ( A 
C_  |^| { x  |  ( x  e.  B  /\  ph ) }  <->  A. x  e.  B  ( ph  ->  A  C_  x )
)
93, 8bitri 240 1  |-  ( A 
C_  |^| { x  e.  B  |  ph }  <->  A. x  e.  B  (
ph  ->  A  C_  x
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   A.wal 1530    e. wcel 1696   {cab 2282   A.wral 2556   {crab 2560    C_ wss 3165   |^|cint 3878
This theorem is referenced by:  knatar  5873  harval2  7646  pwfseqlem3  8298  wuncid  8381  topjoin  26417
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rab 2565  df-v 2803  df-in 3172  df-ss 3179  df-int 3879
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