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Theorem iineq2d 3828
Description: Equality deduction for indexed intersection. (Contributed by NM, 7-Dec-2011.)
Hypotheses
Ref Expression
iineq2d.1  |-  F/ x ph
iineq2d.2  |-  ( (
ph  /\  x  e.  A )  ->  B  =  C )
Assertion
Ref Expression
iineq2d  |-  ( ph  -> 
|^|_ x  e.  A  B  =  |^|_ x  e.  A  C )

Proof of Theorem iineq2d
StepHypRef Expression
1 iineq2d.1 . . 3  |-  F/ x ph
2 iineq2d.2 . . . 4  |-  ( (
ph  /\  x  e.  A )  ->  B  =  C )
32ex 114 . . 3  |-  ( ph  ->  ( x  e.  A  ->  B  =  C ) )
41, 3ralrimi 2501 . 2  |-  ( ph  ->  A. x  e.  A  B  =  C )
5 iineq2 3825 . 2  |-  ( A. x  e.  A  B  =  C  ->  |^|_ x  e.  A  B  =  |^|_
x  e.  A  C
)
64, 5syl 14 1  |-  ( ph  -> 
|^|_ x  e.  A  B  =  |^|_ x  e.  A  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1331   F/wnf 1436    e. wcel 1480   A.wral 2414   |^|_ciin 3809
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-ral 2419  df-iin 3811
This theorem is referenced by:  iineq2dv  3830
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