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Theorem iinssf 45143
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 4996 . . . 4 (𝑦 ∈ V → (𝑦 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴 𝑦𝐵))
21elv 3485 . . 3 (𝑦 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴 𝑦𝐵)
3 ssel 3977 . . . . 5 (𝐵𝐶 → (𝑦𝐵𝑦𝐶))
43reximi 3084 . . . 4 (∃𝑥𝐴 𝐵𝐶 → ∃𝑥𝐴 (𝑦𝐵𝑦𝐶))
5 iinssf.1 . . . . . 6 𝑥𝐶
65nfcri 2897 . . . . 5 𝑥 𝑦𝐶
76r19.36vf 45141 . . . 4 (∃𝑥𝐴 (𝑦𝐵𝑦𝐶) → (∀𝑥𝐴 𝑦𝐵𝑦𝐶))
84, 7syl 17 . . 3 (∃𝑥𝐴 𝐵𝐶 → (∀𝑥𝐴 𝑦𝐵𝑦𝐶))
92, 8biimtrid 242 . 2 (∃𝑥𝐴 𝐵𝐶 → (𝑦 𝑥𝐴 𝐵𝑦𝐶))
109ssrdv 3989 1 (∃𝑥𝐴 𝐵𝐶 𝑥𝐴 𝐵𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wcel 2108  wnfc 2890  wral 3061  wrex 3070  Vcvv 3480  wss 3951   ciin 4992
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-12 2177  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-v 3482  df-ss 3968  df-iin 4994
This theorem is referenced by:  iinssdf  45144  iinss2d  45162
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