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Theorem iineq2dv 4117
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 1630 . 2  |-  F/ x ph
2 iuneq2dv.1 . 2  |-  ( (
ph  /\  x  e.  A )  ->  B  =  C )
31, 2iineq2d 4115 1  |-  ( ph  -> 
|^|_ x  e.  A  B  =  |^|_ x  e.  A  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   |^|_ciin 4096
This theorem is referenced by:  cntziinsn  15138  ptbasfi  17618  fclsval  18045  taylfval  20280  polfvalN  30775  dihglblem3N  32167  dihmeetlem2N  32171
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-ral 2712  df-iin 4098
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