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Theorem iineq2dv 4325
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 1754 . 2  |-  F/ x ph
2 iuneq2dv.1 . 2  |-  ( (
ph  /\  x  e.  A )  ->  B  =  C )
31, 2iineq2d 4323 1  |-  ( ph  -> 
|^|_ x  e.  A  B  =  |^|_ x  e.  A  C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1870   |^|_ciin 4303
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407
This theorem depends on definitions:  df-bi 188  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-clab 2415  df-cleq 2421  df-clel 2424  df-ral 2787  df-iin 4305
This theorem is referenced by:  cntziinsn  16939  ptbasfi  20527  fclsval  20954  taylfval  23179  polfvalN  33178  dihglblem3N  34572  dihmeetlem2N  34576  saliincl  37733
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