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Theorem iineq2 4333
Description: Equality theorem for indexed intersection. (Contributed by NM, 22-Oct-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
Assertion
Ref Expression
iineq2  |-  ( A. x  e.  A  B  =  C  ->  |^|_ x  e.  A  B  =  |^|_
x  e.  A  C
)

Proof of Theorem iineq2
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 eleq2 2527 . . . . 5  |-  ( B  =  C  ->  (
y  e.  B  <->  y  e.  C ) )
21ralimi 2847 . . . 4  |-  ( A. x  e.  A  B  =  C  ->  A. x  e.  A  ( y  e.  B  <->  y  e.  C
) )
3 ralbi 2985 . . . 4  |-  ( A. x  e.  A  (
y  e.  B  <->  y  e.  C )  ->  ( A. x  e.  A  y  e.  B  <->  A. x  e.  A  y  e.  C ) )
42, 3syl 16 . . 3  |-  ( A. x  e.  A  B  =  C  ->  ( A. x  e.  A  y  e.  B  <->  A. x  e.  A  y  e.  C )
)
54abbidv 2590 . 2  |-  ( A. x  e.  A  B  =  C  ->  { y  |  A. x  e.  A  y  e.  B }  =  { y  |  A. x  e.  A  y  e.  C }
)
6 df-iin 4318 . 2  |-  |^|_ x  e.  A  B  =  { y  |  A. x  e.  A  y  e.  B }
7 df-iin 4318 . 2  |-  |^|_ x  e.  A  C  =  { y  |  A. x  e.  A  y  e.  C }
85, 6, 73eqtr4g 2520 1  |-  ( A. x  e.  A  B  =  C  ->  |^|_ x  e.  A  B  =  |^|_
x  e.  A  C
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1398    e. wcel 1823   {cab 2439   A.wral 2804   |^|_ciin 4316
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-an 369  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-ral 2809  df-iin 4318
This theorem is referenced by:  iineq2i  4335  iineq2d  4336  firest  14925  iincld  19710  elrfirn2  30871
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