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Theorem ssintrab 3638
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 2315 . . . 4  |-  { x  e.  B  |  ph }  =  { x  |  ( x  e.  B  /\  ph ) }
21inteqi 3619 . . 3  |-  |^| { x  e.  B  |  ph }  =  |^| { x  |  ( x  e.  B  /\  ph ) }
32sseq2i 2970 . 2  |-  ( A 
C_  |^| { x  e.  B  |  ph }  <->  A 
C_  |^| { x  |  ( x  e.  B  /\  ph ) } )
4 impexp 250 . . . 4  |-  ( ( ( x  e.  B  /\  ph )  ->  A  C_  x )  <->  ( x  e.  B  ->  ( ph  ->  A  C_  x )
) )
54albii 1359 . . 3  |-  ( A. x ( ( x  e.  B  /\  ph )  ->  A  C_  x
)  <->  A. x ( x  e.  B  ->  ( ph  ->  A  C_  x
) ) )
6 ssintab 3632 . . 3  |-  ( A 
C_  |^| { x  |  ( x  e.  B  /\  ph ) }  <->  A. x
( ( x  e.  B  /\  ph )  ->  A  C_  x )
)
7 df-ral 2311 . . 3  |-  ( A. x  e.  B  ( ph  ->  A  C_  x
)  <->  A. x ( x  e.  B  ->  ( ph  ->  A  C_  x
) ) )
85, 6, 73bitr4i 201 . 2  |-  ( A 
C_  |^| { x  |  ( x  e.  B  /\  ph ) }  <->  A. x  e.  B  ( ph  ->  A  C_  x )
)
93, 8bitri 173 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    /\ wa 97    <-> wb 98   A.wal 1241    e. wcel 1393   {cab 2026   A.wral 2306   {crab 2310    C_ wss 2917   |^|cint 3615
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rab 2315  df-v 2559  df-in 2924  df-ss 2931  df-int 3616
This theorem is referenced by: (None)
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