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Theorem eliin 3974
 Description: Membership in indexed intersection. (Contributed by NM, 3-Sep-2003.)
Assertion
Ref Expression
eliin (A V → (A x B Cx B A C))
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 2413 . . 3 (y = A → (y CA C))
21ralbidv 2634 . 2 (y = A → (x B y Cx B A C))
3 df-iin 3972 . 2 x B C = {y x B y C}
42, 3elab2g 2987 1 (A V → (A x B Cx B A C))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   = wceq 1642   ∈ wcel 1710  ∀wral 2614  ∩ciin 3970 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ral 2619  df-v 2861  df-iin 3972 This theorem is referenced by:  iinconst  3978  iuniin  3979  iinss1  3981  ssiinf  4015  iinss  4017  iinss2  4018  iinab  4027  iinun2  4032  iundif2  4033  iindif2  4035  iinin2  4036  elriin  4038  iinxprg  4043  iinuni  4049  iinpw  4054  cnviin  5118
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