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Theorem iinssf 41773
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 4886 . . . 4 (𝑦 ∈ V → (𝑦 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴 𝑦𝐵))
21elv 3446 . . 3 (𝑦 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴 𝑦𝐵)
3 ssel 3908 . . . . 5 (𝐵𝐶 → (𝑦𝐵𝑦𝐶))
43reximi 3206 . . . 4 (∃𝑥𝐴 𝐵𝐶 → ∃𝑥𝐴 (𝑦𝐵𝑦𝐶))
5 iinssf.1 . . . . . 6 𝑥𝐶
65nfcri 2943 . . . . 5 𝑥 𝑦𝐶
76r19.36vf 41770 . . . 4 (∃𝑥𝐴 (𝑦𝐵𝑦𝐶) → (∀𝑥𝐴 𝑦𝐵𝑦𝐶))
84, 7syl 17 . . 3 (∃𝑥𝐴 𝐵𝐶 → (∀𝑥𝐴 𝑦𝐵𝑦𝐶))
92, 8syl5bi 245 . 2 (∃𝑥𝐴 𝐵𝐶 → (𝑦 𝑥𝐴 𝐵𝑦𝐶))
109ssrdv 3921 1 (∃𝑥𝐴 𝐵𝐶 𝑥𝐴 𝐵𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wcel 2111  wnfc 2936  wral 3106  wrex 3107  Vcvv 3441  wss 3881   ciin 4882
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-v 3443  df-in 3888  df-ss 3898  df-iin 4884
This theorem is referenced by:  iinssdf  41774
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