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Theorem iineq2d 3718
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 113 . . 3  |-  ( ph  ->  ( x  e.  A  ->  B  =  C ) )
41, 3ralrimi 2437 . 2  |-  ( ph  ->  A. x  e.  A  B  =  C )
5 iineq2 3715 . 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 102    = wceq 1285   F/wnf 1390    e. wcel 1434   A.wral 2353   |^|_ciin 3699
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-11 1438  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-ral 2358  df-iin 3701
This theorem is referenced by:  iineq2dv  3720
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