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Theorem iinss1 4806
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 3924 . . 3 (𝐴𝐵 → (∀𝑥𝐵 𝑦𝐶 → ∀𝑥𝐴 𝑦𝐶))
2 eliin 4797 . . . 4 (𝑦 ∈ V → (𝑦 𝑥𝐵 𝐶 ↔ ∀𝑥𝐵 𝑦𝐶))
32elv 3421 . . 3 (𝑦 𝑥𝐵 𝐶 ↔ ∀𝑥𝐵 𝑦𝐶)
4 eliin 4797 . . . 4 (𝑦 ∈ V → (𝑦 𝑥𝐴 𝐶 ↔ ∀𝑥𝐴 𝑦𝐶))
54elv 3421 . . 3 (𝑦 𝑥𝐴 𝐶 ↔ ∀𝑥𝐴 𝑦𝐶)
61, 3, 53imtr4g 288 . 2 (𝐴𝐵 → (𝑦 𝑥𝐵 𝐶𝑦 𝑥𝐴 𝐶))
76ssrdv 3865 1 (𝐴𝐵 𝑥𝐵 𝐶 𝑥𝐴 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wcel 2050  wral 3089  Vcvv 3416  wss 3830   ciin 4793
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-ext 2751
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-clab 2760  df-cleq 2772  df-clel 2847  df-nfc 2919  df-ral 3094  df-v 3418  df-in 3837  df-ss 3844  df-iin 4795
This theorem is referenced by:  polcon3N  36495  smflimsuplem5  42527  smflimsuplem7  42529
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