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Theorem iineq2dv 4296
Description: Equality deduction for indexed intersection. (Contributed by NM, 3-Aug-2004.)
Hypothesis
Ref Expression
iuneq2dv.1  |-  ( (
ph  /\  x  e.  A )  ->  B  =  C )
Assertion
Ref Expression
iineq2dv  |-  ( ph  -> 
|^|_ x  e.  A  B  =  |^|_ x  e.  A  C )
Distinct variable group:    ph, x
Allowed substitution hints:    A( x)    B( x)    C( x)

Proof of Theorem iineq2dv
StepHypRef Expression
1 nfv 1674 . 2  |-  F/ x ph
2 iuneq2dv.1 . 2  |-  ( (
ph  /\  x  e.  A )  ->  B  =  C )
31, 2iineq2d 4294 1  |-  ( ph  -> 
|^|_ x  e.  A  B  =  |^|_ x  e.  A  C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   |^|_ciin 4275
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2438  df-cleq 2444  df-clel 2447  df-ral 2801  df-iin 4277
This theorem is referenced by:  cntziinsn  15966  ptbasfi  19281  fclsval  19708  taylfval  21952  polfvalN  33867  dihglblem3N  35259  dihmeetlem2N  35263
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