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Theorem iinss 5060
Description: Subset implication for an indexed intersection. (Contributed by NM, 15-Oct-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
Assertion
Ref Expression
iinss (∃𝑥𝐴 𝐵𝐶 𝑥𝐴 𝐵𝐶)
Distinct variable group:   𝑥,𝐶
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)

Proof of Theorem iinss
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 eliin 5003 . . . 4 (𝑦 ∈ V → (𝑦 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴 𝑦𝐵))
21elv 3481 . . 3 (𝑦 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴 𝑦𝐵)
3 ssel 3976 . . . . 5 (𝐵𝐶 → (𝑦𝐵𝑦𝐶))
43reximi 3085 . . . 4 (∃𝑥𝐴 𝐵𝐶 → ∃𝑥𝐴 (𝑦𝐵𝑦𝐶))
5 r19.36v 3184 . . . 4 (∃𝑥𝐴 (𝑦𝐵𝑦𝐶) → (∀𝑥𝐴 𝑦𝐵𝑦𝐶))
64, 5syl 17 . . 3 (∃𝑥𝐴 𝐵𝐶 → (∀𝑥𝐴 𝑦𝐵𝑦𝐶))
72, 6biimtrid 241 . 2 (∃𝑥𝐴 𝐵𝐶 → (𝑦 𝑥𝐴 𝐵𝑦𝐶))
87ssrdv 3989 1 (∃𝑥𝐴 𝐵𝐶 𝑥𝐴 𝐵𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wcel 2107  wral 3062  wrex 3071  Vcvv 3475  wss 3949   ciin 4999
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-v 3477  df-in 3956  df-ss 3966  df-iin 5001
This theorem is referenced by:  riinn0  5087  reliin  5818  cnviin  6286  iiner  8783  scott0  9881  cfslb  10261  ptbasfi  23085  iscmet3  24810  fnemeet1  35251  pmapglb2N  38642  pmapglb2xN  38643  iinssd  43820  iooiinicc  44255  iooiinioc  44269  meaiininclem  45202  iinhoiicclem  45389  smflim  45493  smflimsuplem7  45542
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