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Theorem iineq2dv 3873
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 1508 . 2 𝑥𝜑
2 iuneq2dv.1 . 2 ((𝜑𝑥𝐴) → 𝐵 = 𝐶)
31, 2iineq2d 3871 1 (𝜑 𝑥𝐴 𝐵 = 𝑥𝐴 𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1335  wcel 2128   ciin 3852
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-11 1486  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-ral 2440  df-iin 3854
This theorem is referenced by: (None)
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