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Theorem iineq2dv 3752
 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 1466 . 2 𝑥𝜑
2 iuneq2dv.1 . 2 ((𝜑𝑥𝐴) → 𝐵 = 𝐶)
31, 2iineq2d 3750 1 (𝜑 𝑥𝐴 𝐵 = 𝑥𝐴 𝐶)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 102   = wceq 1289   ∈ wcel 1438  ∩ ciin 3731 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-11 1442  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070 This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-ral 2364  df-iin 3733 This theorem is referenced by: (None)
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