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Theorem iinssiin 45107
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 . . . . 5 𝑥𝜑
2 nfii1 5027 . . . . . 6 𝑥 𝑥𝐴 𝐵
32nfcri 2896 . . . . 5 𝑥 𝑦 𝑥𝐴 𝐵
41, 3nfan 1899 . . . 4 𝑥(𝜑𝑦 𝑥𝐴 𝐵)
5 iinssiin.2 . . . . . . 7 ((𝜑𝑥𝐴) → 𝐵𝐶)
65adantlr 715 . . . . . 6 (((𝜑𝑦 𝑥𝐴 𝐵) ∧ 𝑥𝐴) → 𝐵𝐶)
7 eliinid 45089 . . . . . . 7 ((𝑦 𝑥𝐴 𝐵𝑥𝐴) → 𝑦𝐵)
87adantll 714 . . . . . 6 (((𝜑𝑦 𝑥𝐴 𝐵) ∧ 𝑥𝐴) → 𝑦𝐵)
96, 8sseldd 3983 . . . . 5 (((𝜑𝑦 𝑥𝐴 𝐵) ∧ 𝑥𝐴) → 𝑦𝐶)
109ex 412 . . . 4 ((𝜑𝑦 𝑥𝐴 𝐵) → (𝑥𝐴𝑦𝐶))
114, 10ralrimi 3256 . . 3 ((𝜑𝑦 𝑥𝐴 𝐵) → ∀𝑥𝐴 𝑦𝐶)
12 eliin 4994 . . . 4 (𝑦 ∈ V → (𝑦 𝑥𝐴 𝐶 ↔ ∀𝑥𝐴 𝑦𝐶))
1312elv 3484 . . 3 (𝑦 𝑥𝐴 𝐶 ↔ ∀𝑥𝐴 𝑦𝐶)
1411, 13sylibr 234 . 2 ((𝜑𝑦 𝑥𝐴 𝐵) → 𝑦 𝑥𝐴 𝐶)
1514ssd 45058 1 (𝜑 𝑥𝐴 𝐵 𝑥𝐴 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wnf 1783  wcel 2108  wral 3060  Vcvv 3479  wss 3950   ciin 4990
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-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ral 3061  df-v 3481  df-ss 3967  df-iin 4992
This theorem is referenced by: (None)
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