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Theorem iinssiin 41549
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 4930 . . . . . 6 𝑥 𝑥𝐴 𝐵
32nfcri 2967 . . . . 5 𝑥 𝑦 𝑥𝐴 𝐵
41, 3nfan 1900 . . . 4 𝑥(𝜑𝑦 𝑥𝐴 𝐵)
5 iinssiin.2 . . . . . . 7 ((𝜑𝑥𝐴) → 𝐵𝐶)
65adantlr 713 . . . . . 6 (((𝜑𝑦 𝑥𝐴 𝐵) ∧ 𝑥𝐴) → 𝐵𝐶)
7 eliinid 41532 . . . . . . 7 ((𝑦 𝑥𝐴 𝐵𝑥𝐴) → 𝑦𝐵)
87adantll 712 . . . . . 6 (((𝜑𝑦 𝑥𝐴 𝐵) ∧ 𝑥𝐴) → 𝑦𝐵)
96, 8sseldd 3947 . . . . 5 (((𝜑𝑦 𝑥𝐴 𝐵) ∧ 𝑥𝐴) → 𝑦𝐶)
109ex 415 . . . 4 ((𝜑𝑦 𝑥𝐴 𝐵) → (𝑥𝐴𝑦𝐶))
114, 10ralrimi 3203 . . 3 ((𝜑𝑦 𝑥𝐴 𝐵) → ∀𝑥𝐴 𝑦𝐶)
12 eliin 4900 . . . 4 (𝑦 ∈ V → (𝑦 𝑥𝐴 𝐶 ↔ ∀𝑥𝐴 𝑦𝐶))
1312elv 3478 . . 3 (𝑦 𝑥𝐴 𝐶 ↔ ∀𝑥𝐴 𝑦𝐶)
1411, 13sylibr 236 . 2 ((𝜑𝑦 𝑥𝐴 𝐵) → 𝑦 𝑥𝐴 𝐶)
1514ssd 41499 1 (𝜑 𝑥𝐴 𝐵 𝑥𝐴 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398  wnf 1784  wcel 2114  wral 3125  Vcvv 3473  wss 3913   ciin 4896
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ral 3130  df-v 3475  df-in 3920  df-ss 3930  df-iin 4898
This theorem is referenced by: (None)
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