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Theorem eliin 3685
 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 2142 . . 3 (𝑦 = 𝐴 → (𝑦𝐶𝐴𝐶))
21ralbidv 2369 . 2 (𝑦 = 𝐴 → (∀𝑥𝐵 𝑦𝐶 ↔ ∀𝑥𝐵 𝐴𝐶))
3 df-iin 3683 . 2 𝑥𝐵 𝐶 = {𝑦 ∣ ∀𝑥𝐵 𝑦𝐶}
42, 3elab2g 2741 1 (𝐴𝑉 → (𝐴 𝑥𝐵 𝐶 ↔ ∀𝑥𝐵 𝐴𝐶))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 103   = wceq 1285   ∈ wcel 1434  ∀wral 2349  ∩ ciin 3681 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-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064 This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-v 2604  df-iin 3683 This theorem is referenced by:  iinconstm  3689  iuniin  3690  iinss1  3692  ssiinf  3729  iinss  3731  iinss2  3732  iinab  3741  iundif2ss  3745  iindif2m  3747  iinin2m  3748  elriin  3750  iinpw  3765  xpiindim  4495  cnviinm  4883  iinerm  6237
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