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Theorem iineq2dv 4479
 Description: Equality deduction for indexed intersection. (Contributed by NM, 3-Aug-2004.)
Hypothesis
Ref Expression
iuneq2dv.1 ((𝜑𝑥𝐴) → 𝐵 = 𝐶)
Assertion
Ref Expression
iineq2dv (𝜑 𝑥𝐴 𝐵 = 𝑥𝐴 𝐶)
Distinct variable group:   𝜑,𝑥
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)   𝐶(𝑥)

Proof of Theorem iineq2dv
StepHypRef Expression
1 nfv 1830 . 2 𝑥𝜑
2 iuneq2dv.1 . 2 ((𝜑𝑥𝐴) → 𝐵 = 𝐶)
31, 2iineq2d 4477 1 (𝜑 𝑥𝐴 𝐵 = 𝑥𝐴 𝐶)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∩ ciin 4456 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-ral 2901  df-iin 4458 This theorem is referenced by:  cntziinsn  17590  ptbasfi  21194  fclsval  21622  taylfval  23917  polfvalN  34208  dihglblem3N  35602  dihmeetlem2N  35606  iineq12dv  38319  saliincl  39221  iccvonmbllem  39569  vonicclem2  39575  smflimlem3  39659  smflimlem4  39660  smflimlem6  39662
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