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Theorem ssintrab 4065
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 2706 . . . 4  |-  { x  e.  B  |  ph }  =  { x  |  ( x  e.  B  /\  ph ) }
21inteqi 4046 . . 3  |-  |^| { x  e.  B  |  ph }  =  |^| { x  |  ( x  e.  B  /\  ph ) }
32sseq2i 3365 . 2  |-  ( A 
C_  |^| { x  e.  B  |  ph }  <->  A 
C_  |^| { x  |  ( x  e.  B  /\  ph ) } )
4 impexp 434 . . . 4  |-  ( ( ( x  e.  B  /\  ph )  ->  A  C_  x )  <->  ( x  e.  B  ->  ( ph  ->  A  C_  x )
) )
54albii 1575 . . 3  |-  ( A. x ( ( x  e.  B  /\  ph )  ->  A  C_  x
)  <->  A. x ( x  e.  B  ->  ( ph  ->  A  C_  x
) ) )
6 ssintab 4059 . . 3  |-  ( A 
C_  |^| { x  |  ( x  e.  B  /\  ph ) }  <->  A. x
( ( x  e.  B  /\  ph )  ->  A  C_  x )
)
7 df-ral 2702 . . 3  |-  ( A. x  e.  B  ( ph  ->  A  C_  x
)  <->  A. x ( x  e.  B  ->  ( ph  ->  A  C_  x
) ) )
85, 6, 73bitr4i 269 . 2  |-  ( A 
C_  |^| { x  |  ( x  e.  B  /\  ph ) }  <->  A. x  e.  B  ( ph  ->  A  C_  x )
)
93, 8bitri 241 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 177    /\ wa 359   A.wal 1549    e. wcel 1725   {cab 2421   A.wral 2697   {crab 2701    C_ wss 3312   |^|cint 4042
This theorem is referenced by:  knatar  6072  harval2  7876  pwfseqlem3  8527  topjoin  26385
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702  df-rab 2706  df-v 2950  df-in 3319  df-ss 3326  df-int 4043
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