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Theorem iinrabm 3747
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 3342 . . 3  |-  ( E. x  x  e.  A  ->  ( A. x  e.  A  ( y  e.  B  /\  ph )  <->  ( y  e.  B  /\  A. x  e.  A  ph ) ) )
21abbidv 2171 . 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 2332 . . . . 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 3704 . . 3  |-  |^|_ x  e.  A  { y  e.  B  |  ph }  =  |^|_ x  e.  A  { y  |  ( y  e.  B  /\  ph ) }
6 iinab 3746 . . 3  |-  |^|_ x  e.  A  { y  |  ( y  e.  B  /\  ph ) }  =  { y  |  A. x  e.  A  ( y  e.  B  /\  ph ) }
75, 6eqtri 2076 . 2  |-  |^|_ x  e.  A  { y  e.  B  |  ph }  =  { y  |  A. x  e.  A  (
y  e.  B  /\  ph ) }
8 df-rab 2332 . 2  |-  { y  e.  B  |  A. x  e.  A  ph }  =  { y  |  ( y  e.  B  /\  A. x  e.  A  ph ) }
92, 7, 83eqtr4g 2113 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 101    = wceq 1259   E.wex 1397    e. wcel 1409   {cab 2042   A.wral 2323   {crab 2327   |^|_ciin 3686
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rab 2332  df-v 2576  df-iin 3688
This theorem is referenced by: (None)
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