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Theorem elintrab 4064
Description: Membership in the intersection of a class abstraction. (Contributed by NM, 17-Oct-1999.)
Hypothesis
Ref Expression
inteqab.1  |-  A  e. 
_V
Assertion
Ref Expression
elintrab  |-  ( A  e.  |^| { x  e.  B  |  ph }  <->  A. x  e.  B  (
ph  ->  A  e.  x
) )
Distinct variable group:    x, A
Allowed substitution hints:    ph( x)    B( x)

Proof of Theorem elintrab
StepHypRef Expression
1 inteqab.1 . . . 4  |-  A  e. 
_V
21elintab 4063 . . 3  |-  ( A  e.  |^| { x  |  ( x  e.  B  /\  ph ) }  <->  A. x
( ( x  e.  B  /\  ph )  ->  A  e.  x ) )
3 impexp 435 . . . 4  |-  ( ( ( x  e.  B  /\  ph )  ->  A  e.  x )  <->  ( x  e.  B  ->  ( ph  ->  A  e.  x ) ) )
43albii 1576 . . 3  |-  ( A. x ( ( x  e.  B  /\  ph )  ->  A  e.  x
)  <->  A. x ( x  e.  B  ->  ( ph  ->  A  e.  x
) ) )
52, 4bitri 242 . 2  |-  ( A  e.  |^| { x  |  ( x  e.  B  /\  ph ) }  <->  A. x
( x  e.  B  ->  ( ph  ->  A  e.  x ) ) )
6 df-rab 2716 . . . 4  |-  { x  e.  B  |  ph }  =  { x  |  ( x  e.  B  /\  ph ) }
76inteqi 4056 . . 3  |-  |^| { x  e.  B  |  ph }  =  |^| { x  |  ( x  e.  B  /\  ph ) }
87eleq2i 2502 . 2  |-  ( A  e.  |^| { x  e.  B  |  ph }  <->  A  e.  |^| { x  |  ( x  e.  B  /\  ph ) } )
9 df-ral 2712 . 2  |-  ( A. x  e.  B  ( ph  ->  A  e.  x
)  <->  A. x ( x  e.  B  ->  ( ph  ->  A  e.  x
) ) )
105, 8, 93bitr4i 270 1  |-  ( A  e.  |^| { x  e.  B  |  ph }  <->  A. x  e.  B  (
ph  ->  A  e.  x
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360   A.wal 1550    e. wcel 1726   {cab 2424   A.wral 2707   {crab 2711   _Vcvv 2958   |^|cint 4052
This theorem is referenced by:  elintrabg  4065  intmin  4072  rankunb  7778  isf34lem4  8259  ist1-3  17415  filufint  17954  elspani  23047  kur14lem9  24902  pclclN  30690  elpclN  30691
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-rab 2716  df-v 2960  df-int 4053
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