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Theorem iineq2d 3941
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 423 . . 3  |-  ( ph  ->  ( x  e.  A  ->  B  =  C ) )
41, 3ralrimi 2637 . 2  |-  ( ph  ->  A. x  e.  A  B  =  C )
5 iineq2 3938 . 2  |-  ( A. x  e.  A  B  =  C  ->  |^|_ x  e.  A  B  =  |^|_
x  e.  A  C
)
64, 5syl 15 1  |-  ( ph  -> 
|^|_ x  e.  A  B  =  |^|_ x  e.  A  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   F/wnf 1534    = wceq 1632    e. wcel 1696   A.wral 2556   |^|_ciin 3922
This theorem is referenced by:  iineq2dv  3943  pmapglbx  30580
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-ral 2561  df-iin 3924
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