ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  eliin GIF version

Theorem eliin 3905
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 2251 . . 3 (𝑦 = 𝐴 → (𝑦𝐶𝐴𝐶))
21ralbidv 2489 . 2 (𝑦 = 𝐴 → (∀𝑥𝐵 𝑦𝐶 ↔ ∀𝑥𝐵 𝐴𝐶))
3 df-iin 3903 . 2 𝑥𝐵 𝐶 = {𝑦 ∣ ∀𝑥𝐵 𝑦𝐶}
42, 3elab2g 2898 1 (𝐴𝑉 → (𝐴 𝑥𝐵 𝐶 ↔ ∀𝑥𝐵 𝐴𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105   = wceq 1363  wcel 2159  wral 2467   ciin 3901
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2170
This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2175  df-cleq 2181  df-clel 2184  df-nfc 2320  df-ral 2472  df-v 2753  df-iin 3903
This theorem is referenced by:  iinconstm  3909  iuniin  3910  iinss1  3912  ssiinf  3950  iinss  3952  iinss2  3953  iinab  3962  iundif2ss  3966  iindif2m  3968  iinin2m  3969  elriin  3971  iinpw  3991  xpiindim  4778  cnviinm  5184  iinerm  6624  ixpiinm  6741
  Copyright terms: Public domain W3C validator