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Theorem iinssiin 39132
Description: Subset implication for an indexed intersection. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
Hypotheses
Ref Expression
iinssiin.1 𝑥𝜑
iinssiin.2 ((𝜑𝑥𝐴) → 𝐵𝐶)
Assertion
Ref Expression
iinssiin (𝜑 𝑥𝐴 𝐵 𝑥𝐴 𝐶)

Proof of Theorem iinssiin
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 iinssiin.1 . . . . . 6 𝑥𝜑
2 nfcv 2762 . . . . . . 7 𝑥𝑦
3 nfii1 4542 . . . . . . 7 𝑥 𝑥𝐴 𝐵
42, 3nfel 2774 . . . . . 6 𝑥 𝑦 𝑥𝐴 𝐵
51, 4nfan 1826 . . . . 5 𝑥(𝜑𝑦 𝑥𝐴 𝐵)
6 iinssiin.2 . . . . . . . 8 ((𝜑𝑥𝐴) → 𝐵𝐶)
76adantlr 750 . . . . . . 7 (((𝜑𝑦 𝑥𝐴 𝐵) ∧ 𝑥𝐴) → 𝐵𝐶)
8 eliinid 39114 . . . . . . . 8 ((𝑦 𝑥𝐴 𝐵𝑥𝐴) → 𝑦𝐵)
98adantll 749 . . . . . . 7 (((𝜑𝑦 𝑥𝐴 𝐵) ∧ 𝑥𝐴) → 𝑦𝐵)
107, 9sseldd 3596 . . . . . 6 (((𝜑𝑦 𝑥𝐴 𝐵) ∧ 𝑥𝐴) → 𝑦𝐶)
1110ex 450 . . . . 5 ((𝜑𝑦 𝑥𝐴 𝐵) → (𝑥𝐴𝑦𝐶))
125, 11ralrimi 2954 . . . 4 ((𝜑𝑦 𝑥𝐴 𝐵) → ∀𝑥𝐴 𝑦𝐶)
13 vex 3198 . . . . 5 𝑦 ∈ V
14 eliin 4516 . . . . 5 (𝑦 ∈ V → (𝑦 𝑥𝐴 𝐶 ↔ ∀𝑥𝐴 𝑦𝐶))
1513, 14ax-mp 5 . . . 4 (𝑦 𝑥𝐴 𝐶 ↔ ∀𝑥𝐴 𝑦𝐶)
1612, 15sylibr 224 . . 3 ((𝜑𝑦 𝑥𝐴 𝐵) → 𝑦 𝑥𝐴 𝐶)
1716ralrimiva 2963 . 2 (𝜑 → ∀𝑦 𝑥𝐴 𝐵𝑦 𝑥𝐴 𝐶)
18 dfss3 3585 . 2 ( 𝑥𝐴 𝐵 𝑥𝐴 𝐶 ↔ ∀𝑦 𝑥𝐴 𝐵𝑦 𝑥𝐴 𝐶)
1917, 18sylibr 224 1 (𝜑 𝑥𝐴 𝐵 𝑥𝐴 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  wnf 1706  wcel 1988  wral 2909  Vcvv 3195  wss 3567   ciin 4512
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-v 3197  df-in 3574  df-ss 3581  df-iin 4514
This theorem is referenced by: (None)
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