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Theorem iineq1 4296
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 3023 . . 3  |-  ( A  =  B  ->  ( A. x  e.  A  y  e.  C  <->  A. x  e.  B  y  e.  C ) )
21abbidv 2590 . 2  |-  ( A  =  B  ->  { y  |  A. x  e.  A  y  e.  C }  =  { y  |  A. x  e.  B  y  e.  C }
)
3 df-iin 4285 . 2  |-  |^|_ x  e.  A  C  =  { y  |  A. x  e.  A  y  e.  C }
4 df-iin 4285 . 2  |-  |^|_ x  e.  B  C  =  { y  |  A. x  e.  B  y  e.  C }
52, 3, 43eqtr4g 2520 1  |-  ( A  =  B  ->  |^|_ x  e.  A  C  =  |^|_
x  e.  B  C
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    e. wcel 1758   {cab 2439   A.wral 2799   |^|_ciin 4283
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ral 2804  df-iin 4285
This theorem is referenced by:  iinrab2  4344  iinvdif  4353  riin0  4355  iin0  4577  xpriindi  5087  cmpfi  19153  ptbasfi  19296  fclsval  19723  taylfval  21967  polvalN  33912
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