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Theorem iinssf 41283
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 4915 . . . 4 (𝑦 ∈ V → (𝑦 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴 𝑦𝐵))
21elv 3497 . . 3 (𝑦 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴 𝑦𝐵)
3 ssel 3958 . . . . 5 (𝐵𝐶 → (𝑦𝐵𝑦𝐶))
43reximi 3240 . . . 4 (∃𝑥𝐴 𝐵𝐶 → ∃𝑥𝐴 (𝑦𝐵𝑦𝐶))
5 iinssf.1 . . . . . 6 𝑥𝐶
65nfcri 2968 . . . . 5 𝑥 𝑦𝐶
76r19.36vf 41280 . . . 4 (∃𝑥𝐴 (𝑦𝐵𝑦𝐶) → (∀𝑥𝐴 𝑦𝐵𝑦𝐶))
84, 7syl 17 . . 3 (∃𝑥𝐴 𝐵𝐶 → (∀𝑥𝐴 𝑦𝐵𝑦𝐶))
92, 8syl5bi 243 . 2 (∃𝑥𝐴 𝐵𝐶 → (𝑦 𝑥𝐴 𝐵𝑦𝐶))
109ssrdv 3970 1 (∃𝑥𝐴 𝐵𝐶 𝑥𝐴 𝐵𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wcel 2105  wnfc 2958  wral 3135  wrex 3136  Vcvv 3492  wss 3933   ciin 4911
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-v 3494  df-in 3940  df-ss 3949  df-iin 4913
This theorem is referenced by:  iinssdf  41284
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