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Theorem iinssf 45078
Description: Subset implication for an indexed intersection. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
Hypothesis
Ref Expression
iinssf.1 𝑥𝐶
Assertion
Ref Expression
iinssf (∃𝑥𝐴 𝐵𝐶 𝑥𝐴 𝐵𝐶)

Proof of Theorem iinssf
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 iinssf.1 . . . . . 6 𝑥𝐶
65nfcri 2895 . . . . 5 𝑥 𝑦𝐶
76r19.36vf 45076 . . . 4 (∃𝑥𝐴 (𝑦𝐵𝑦𝐶) → (∀𝑥𝐴 𝑦𝐵𝑦𝐶))
84, 7syl 17 . . 3 (∃𝑥𝐴 𝐵𝐶 → (∀𝑥𝐴 𝑦𝐵𝑦𝐶))
92, 8biimtrid 242 . 2 (∃𝑥𝐴 𝐵𝐶 → (𝑦 𝑥𝐴 𝐵𝑦𝐶))
109ssrdv 4001 1 (∃𝑥𝐴 𝐵𝐶 𝑥𝐴 𝐵𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wcel 2106  wnfc 2888  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-12 2175  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-v 3480  df-ss 3980  df-iin 4999
This theorem is referenced by:  iinssdf  45079  iinss2d  45100
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