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Theorem elintrab 4259
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 4258 . . 3  |-  ( A  e.  |^| { x  |  ( x  e.  B  /\  ph ) }  <->  A. x
( ( x  e.  B  /\  ph )  ->  A  e.  x ) )
3 impexp 452 . . . 4  |-  ( ( ( x  e.  B  /\  ph )  ->  A  e.  x )  <->  ( x  e.  B  ->  ( ph  ->  A  e.  x ) ) )
43albii 1701 . . 3  |-  ( A. x ( ( x  e.  B  /\  ph )  ->  A  e.  x
)  <->  A. x ( x  e.  B  ->  ( ph  ->  A  e.  x
) ) )
52, 4bitri 257 . 2  |-  ( A  e.  |^| { x  |  ( x  e.  B  /\  ph ) }  <->  A. x
( x  e.  B  ->  ( ph  ->  A  e.  x ) ) )
6 df-rab 2757 . . . 4  |-  { x  e.  B  |  ph }  =  { x  |  ( x  e.  B  /\  ph ) }
76inteqi 4251 . . 3  |-  |^| { x  e.  B  |  ph }  =  |^| { x  |  ( x  e.  B  /\  ph ) }
87eleq2i 2531 . 2  |-  ( A  e.  |^| { x  e.  B  |  ph }  <->  A  e.  |^| { x  |  ( x  e.  B  /\  ph ) } )
9 df-ral 2753 . 2  |-  ( A. x  e.  B  ( ph  ->  A  e.  x
)  <->  A. x ( x  e.  B  ->  ( ph  ->  A  e.  x
) ) )
105, 8, 93bitr4i 285 1  |-  ( A  e.  |^| { x  e.  B  |  ph }  <->  A. x  e.  B  (
ph  ->  A  e.  x
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 375   A.wal 1452    e. wcel 1897   {cab 2447   A.wral 2748   {crab 2752   _Vcvv 3056   |^|cint 4247
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441
This theorem depends on definitions:  df-bi 190  df-an 377  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ral 2753  df-rab 2757  df-v 3058  df-int 4248
This theorem is referenced by:  elintrabg  4260  intmin  4267  rankunb  8346  isf34lem4  8832  ist1-3  20413  filufint  20983  elspani  27244  ldsysgenld  29030  ldgenpisyslem1  29033  kur14lem9  29985  pclclN  33500  elpclN  33501  lcosslsp  40503
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