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Theorem iinss1 5011
Description: Subclass theorem for indexed intersection. (Contributed by NM, 24-Jan-2012.)
Assertion
Ref Expression
iinss1 (𝐴𝐵 𝑥𝐵 𝐶 𝑥𝐴 𝐶)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hint:   𝐶(𝑥)

Proof of Theorem iinss1
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 ssralv 4048 . . 3 (𝐴𝐵 → (∀𝑥𝐵 𝑦𝐶 → ∀𝑥𝐴 𝑦𝐶))
2 eliin 5001 . . . 4 (𝑦 ∈ V → (𝑦 𝑥𝐵 𝐶 ↔ ∀𝑥𝐵 𝑦𝐶))
32elv 3477 . . 3 (𝑦 𝑥𝐵 𝐶 ↔ ∀𝑥𝐵 𝑦𝐶)
4 eliin 5001 . . . 4 (𝑦 ∈ V → (𝑦 𝑥𝐴 𝐶 ↔ ∀𝑥𝐴 𝑦𝐶))
54elv 3477 . . 3 (𝑦 𝑥𝐴 𝐶 ↔ ∀𝑥𝐴 𝑦𝐶)
61, 3, 53imtr4g 296 . 2 (𝐴𝐵 → (𝑦 𝑥𝐵 𝐶𝑦 𝑥𝐴 𝐶))
76ssrdv 3986 1 (𝐴𝐵 𝑥𝐵 𝐶 𝑥𝐴 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wcel 2099  wral 3058  Vcvv 3471  wss 3947   ciin 4997
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-v 3473  df-in 3954  df-ss 3964  df-iin 4999
This theorem is referenced by:  polcon3N  39390  smflimsuplem5  46212  smflimsuplem7  46214
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