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Theorem iinssiin 44119
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 5031 . . . . . 6 𝑥 𝑥𝐴 𝐵
32nfcri 2888 . . . . 5 𝑥 𝑦 𝑥𝐴 𝐵
41, 3nfan 1900 . . . 4 𝑥(𝜑𝑦 𝑥𝐴 𝐵)
5 iinssiin.2 . . . . . . 7 ((𝜑𝑥𝐴) → 𝐵𝐶)
65adantlr 711 . . . . . 6 (((𝜑𝑦 𝑥𝐴 𝐵) ∧ 𝑥𝐴) → 𝐵𝐶)
7 eliinid 44101 . . . . . . 7 ((𝑦 𝑥𝐴 𝐵𝑥𝐴) → 𝑦𝐵)
87adantll 710 . . . . . 6 (((𝜑𝑦 𝑥𝐴 𝐵) ∧ 𝑥𝐴) → 𝑦𝐵)
96, 8sseldd 3982 . . . . 5 (((𝜑𝑦 𝑥𝐴 𝐵) ∧ 𝑥𝐴) → 𝑦𝐶)
109ex 411 . . . 4 ((𝜑𝑦 𝑥𝐴 𝐵) → (𝑥𝐴𝑦𝐶))
114, 10ralrimi 3252 . . 3 ((𝜑𝑦 𝑥𝐴 𝐵) → ∀𝑥𝐴 𝑦𝐶)
12 eliin 5001 . . . 4 (𝑦 ∈ V → (𝑦 𝑥𝐴 𝐶 ↔ ∀𝑥𝐴 𝑦𝐶))
1312elv 3478 . . 3 (𝑦 𝑥𝐴 𝐶 ↔ ∀𝑥𝐴 𝑦𝐶)
1411, 13sylibr 233 . 2 ((𝜑𝑦 𝑥𝐴 𝐵) → 𝑦 𝑥𝐴 𝐶)
1514ssd 44070 1 (𝜑 𝑥𝐴 𝐵 𝑥𝐴 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  wnf 1783  wcel 2104  wral 3059  Vcvv 3472  wss 3947   ciin 4997
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-tru 1542  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ral 3060  df-v 3474  df-in 3954  df-ss 3964  df-iin 4999
This theorem is referenced by: (None)
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