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Theorem iineq2dv 4511
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 1840 . 2 𝑥𝜑
2 iuneq2dv.1 . 2 ((𝜑𝑥𝐴) → 𝐵 = 𝐶)
31, 2iineq2d 4509 1 (𝜑 𝑥𝐴 𝐵 = 𝑥𝐴 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987   ciin 4488
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-ral 2912  df-iin 4490
This theorem is referenced by:  cntziinsn  17691  ptbasfi  21297  fclsval  21725  taylfval  24024  polfvalN  34691  dihglblem3N  36085  dihmeetlem2N  36089  iineq12dv  38794  saliincl  39868  iccvonmbllem  40215  vonicclem2  40221  smflimlem3  40304  smflimlem4  40305  smflimlem6  40307  smflimsuplem3  40351
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