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Theorem ssintrab 4016
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 2659 . . . 4  |-  { x  e.  B  |  ph }  =  { x  |  ( x  e.  B  /\  ph ) }
21inteqi 3997 . . 3  |-  |^| { x  e.  B  |  ph }  =  |^| { x  |  ( x  e.  B  /\  ph ) }
32sseq2i 3317 . 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 1572 . . 3  |-  ( A. x ( ( x  e.  B  /\  ph )  ->  A  C_  x
)  <->  A. x ( x  e.  B  ->  ( ph  ->  A  C_  x
) ) )
6 ssintab 4010 . . 3  |-  ( A 
C_  |^| { x  |  ( x  e.  B  /\  ph ) }  <->  A. x
( ( x  e.  B  /\  ph )  ->  A  C_  x )
)
7 df-ral 2655 . . 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 1546    e. wcel 1717   {cab 2374   A.wral 2650   {crab 2654    C_ wss 3264   |^|cint 3993
This theorem is referenced by:  knatar  6020  harval2  7818  pwfseqlem3  8469  topjoin  26086
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 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ral 2655  df-rab 2659  df-v 2902  df-in 3271  df-ss 3278  df-int 3994
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