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Theorem iinssf 39641
 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 vex 3234 . . . 4 𝑦 ∈ V
2 eliin 4557 . . . 4 (𝑦 ∈ V → (𝑦 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴 𝑦𝐵))
31, 2ax-mp 5 . . 3 (𝑦 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴 𝑦𝐵)
4 ssel 3630 . . . . 5 (𝐵𝐶 → (𝑦𝐵𝑦𝐶))
54reximi 3040 . . . 4 (∃𝑥𝐴 𝐵𝐶 → ∃𝑥𝐴 (𝑦𝐵𝑦𝐶))
6 nfcv 2793 . . . . . 6 𝑥𝑦
7 iinssf.1 . . . . . 6 𝑥𝐶
86, 7nfel 2806 . . . . 5 𝑥 𝑦𝐶
98r19.36vf 39638 . . . 4 (∃𝑥𝐴 (𝑦𝐵𝑦𝐶) → (∀𝑥𝐴 𝑦𝐵𝑦𝐶))
105, 9syl 17 . . 3 (∃𝑥𝐴 𝐵𝐶 → (∀𝑥𝐴 𝑦𝐵𝑦𝐶))
113, 10syl5bi 232 . 2 (∃𝑥𝐴 𝐵𝐶 → (𝑦 𝑥𝐴 𝐵𝑦𝐶))
1211ssrdv 3642 1 (∃𝑥𝐴 𝐵𝐶 𝑥𝐴 𝐵𝐶)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∈ wcel 2030  Ⅎwnfc 2780  ∀wral 2941  ∃wrex 2942  Vcvv 3231   ⊆ wss 3607  ∩ ciin 4553 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-v 3233  df-in 3614  df-ss 3621  df-iin 4555 This theorem is referenced by:  iinssdf  39642
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