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Theorem iineq1 4307
Description: Equality theorem for indexed intersection. (Contributed by NM, 27-Jun-1998.)
Assertion
Ref Expression
iineq1  |-  ( A  =  B  ->  |^|_ x  e.  A  C  =  |^|_
x  e.  B  C
)
Distinct variable groups:    x, A    x, B
Allowed substitution hint:    C( x)

Proof of Theorem iineq1
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 raleq 2999 . . 3  |-  ( A  =  B  ->  ( A. x  e.  A  y  e.  C  <->  A. x  e.  B  y  e.  C ) )
21abbidv 2580 . 2  |-  ( A  =  B  ->  { y  |  A. x  e.  A  y  e.  C }  =  { y  |  A. x  e.  B  y  e.  C }
)
3 df-iin 4295 . 2  |-  |^|_ x  e.  A  C  =  { y  |  A. x  e.  A  y  e.  C }
4 df-iin 4295 . 2  |-  |^|_ x  e.  B  C  =  { y  |  A. x  e.  B  y  e.  C }
52, 3, 43eqtr4g 2521 1  |-  ( A  =  B  ->  |^|_ x  e.  A  C  =  |^|_
x  e.  B  C
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1455    e. wcel 1898   {cab 2448   A.wral 2749   |^|_ciin 4293
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442
This theorem depends on definitions:  df-bi 190  df-an 377  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ral 2754  df-iin 4295
This theorem is referenced by:  iinrab2  4355  iinvdif  4364  riin0  4366  iin0  4591  xpriindi  4990  cmpfi  20472  ptbasfi  20645  fclsval  21072  taylfval  23363  polvalN  33515
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