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Theorem iineq2dv 4079
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 1626 . 2  |-  F/ x ph
2 iuneq2dv.1 . 2  |-  ( (
ph  /\  x  e.  A )  ->  B  =  C )
31, 2iineq2d 4077 1  |-  ( ph  -> 
|^|_ x  e.  A  B  =  |^|_ x  e.  A  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   |^|_ciin 4058
This theorem is referenced by:  cntziinsn  15092  ptbasfi  17570  fclsval  17997  taylfval  20232  polfvalN  30390  dihglblem3N  31782  dihmeetlem2N  31786
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2395  df-cleq 2401  df-clel 2404  df-ral 2675  df-iin 4060
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