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Theorem iinrabm 3822
Description: Indexed intersection of a restricted class builder. (Contributed by Jim Kingdon, 16-Aug-2018.)
Assertion
Ref Expression
iinrabm  |-  ( E. x  x  e.  A  -> 
|^|_ x  e.  A  { y  e.  B  |  ph }  =  {
y  e.  B  |  A. x  e.  A  ph } )
Distinct variable groups:    y, A, x   
x, B
Allowed substitution hints:    ph( x, y)    B( y)

Proof of Theorem iinrabm
StepHypRef Expression
1 r19.28mv 3402 . . 3  |-  ( E. x  x  e.  A  ->  ( A. x  e.  A  ( y  e.  B  /\  ph )  <->  ( y  e.  B  /\  A. x  e.  A  ph ) ) )
21abbidv 2217 . 2  |-  ( E. x  x  e.  A  ->  { y  |  A. x  e.  A  (
y  e.  B  /\  ph ) }  =  {
y  |  ( y  e.  B  /\  A. x  e.  A  ph ) } )
3 df-rab 2384 . . . . 5  |-  { y  e.  B  |  ph }  =  { y  |  ( y  e.  B  /\  ph ) }
43a1i 9 . . . 4  |-  ( x  e.  A  ->  { y  e.  B  |  ph }  =  { y  |  ( y  e.  B  /\  ph ) } )
54iineq2i 3779 . . 3  |-  |^|_ x  e.  A  { y  e.  B  |  ph }  =  |^|_ x  e.  A  { y  |  ( y  e.  B  /\  ph ) }
6 iinab 3821 . . 3  |-  |^|_ x  e.  A  { y  |  ( y  e.  B  /\  ph ) }  =  { y  |  A. x  e.  A  ( y  e.  B  /\  ph ) }
75, 6eqtri 2120 . 2  |-  |^|_ x  e.  A  { y  e.  B  |  ph }  =  { y  |  A. x  e.  A  (
y  e.  B  /\  ph ) }
8 df-rab 2384 . 2  |-  { y  e.  B  |  A. x  e.  A  ph }  =  { y  |  ( y  e.  B  /\  A. x  e.  A  ph ) }
92, 7, 83eqtr4g 2157 1  |-  ( E. x  x  e.  A  -> 
|^|_ x  e.  A  { y  e.  B  |  ph }  =  {
y  e.  B  |  A. x  e.  A  ph } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1299   E.wex 1436    e. wcel 1448   {cab 2086   A.wral 2375   {crab 2379   |^|_ciin 3761
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082
This theorem depends on definitions:  df-bi 116  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-rab 2384  df-v 2643  df-iin 3763
This theorem is referenced by: (None)
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