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Theorem iineq2d 3677
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 108 . . 3  |-  ( ph  ->  ( x  e.  A  ->  B  =  C ) )
41, 3ralrimi 2390 . 2  |-  ( ph  ->  A. x  e.  A  B  =  C )
5 iineq2 3674 . 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 97    = wceq 1243   F/wnf 1349    e. wcel 1393   A.wral 2306   |^|_ciin 3658
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-ral 2311  df-iin 3660
This theorem is referenced by:  iineq2dv  3679
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