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Theorem iinss 5061
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 5001 . . . 4 (𝑦 ∈ V → (𝑦 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴 𝑦𝐵))
21elv 3483 . . 3 (𝑦 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴 𝑦𝐵)
3 ssel 3989 . . . . 5 (𝐵𝐶 → (𝑦𝐵𝑦𝐶))
43reximi 3082 . . . 4 (∃𝑥𝐴 𝐵𝐶 → ∃𝑥𝐴 (𝑦𝐵𝑦𝐶))
5 r19.36v 3182 . . . 4 (∃𝑥𝐴 (𝑦𝐵𝑦𝐶) → (∀𝑥𝐴 𝑦𝐵𝑦𝐶))
64, 5syl 17 . . 3 (∃𝑥𝐴 𝐵𝐶 → (∀𝑥𝐴 𝑦𝐵𝑦𝐶))
72, 6biimtrid 242 . 2 (∃𝑥𝐴 𝐵𝐶 → (𝑦 𝑥𝐴 𝐵𝑦𝐶))
87ssrdv 4001 1 (∃𝑥𝐴 𝐵𝐶 𝑥𝐴 𝐵𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wcel 2106  wral 3059  wrex 3068  Vcvv 3478  wss 3963   ciin 4997
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-v 3480  df-ss 3980  df-iin 4999
This theorem is referenced by:  riinn0  5088  reliin  5830  cnviin  6308  iiner  8828  scott0  9924  cfslb  10304  ptbasfi  23605  iscmet3  25341  fnemeet1  36349  pmapglb2N  39754  pmapglb2xN  39755  iinssd  45071  iooiinicc  45495  iooiinioc  45509  meaiininclem  46442  iinhoiicclem  46629  smflim  46733  smflimsuplem7  46782
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