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Theorem eliin 3822
 Description: Membership in indexed intersection. (Contributed by NM, 3-Sep-2003.)
Assertion
Ref Expression
eliin (𝐴𝑉 → (𝐴 𝑥𝐵 𝐶 ↔ ∀𝑥𝐵 𝐴𝐶))
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑥)   𝑉(𝑥)

Proof of Theorem eliin
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 eleq1 2203 . . 3 (𝑦 = 𝐴 → (𝑦𝐶𝐴𝐶))
21ralbidv 2438 . 2 (𝑦 = 𝐴 → (∀𝑥𝐵 𝑦𝐶 ↔ ∀𝑥𝐵 𝐴𝐶))
3 df-iin 3820 . 2 𝑥𝐵 𝐶 = {𝑦 ∣ ∀𝑥𝐵 𝑦𝐶}
42, 3elab2g 2832 1 (𝐴𝑉 → (𝐴 𝑥𝐵 𝐶 ↔ ∀𝑥𝐵 𝐴𝐶))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 104   = wceq 1332   ∈ wcel 1481  ∀wral 2417  ∩ ciin 3818 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-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122 This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-v 2689  df-iin 3820 This theorem is referenced by:  iinconstm  3826  iuniin  3827  iinss1  3829  ssiinf  3866  iinss  3868  iinss2  3869  iinab  3878  iundif2ss  3882  iindif2m  3884  iinin2m  3885  elriin  3887  iinpw  3907  xpiindim  4680  cnviinm  5084  iinerm  6505  ixpiinm  6622
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