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Theorem viin 4848
Description: Indexed intersection with a universal index class. When 𝐴 doesn't depend on 𝑥, this evaluates to 𝐴 by 19.3 2131 and abid2 2903. When 𝐴 = 𝑥, this evaluates to by intiin 4843 and intv 5111. (Contributed by NM, 11-Sep-2008.)
Assertion
Ref Expression
viin 𝑥 ∈ V 𝐴 = {𝑦 ∣ ∀𝑥 𝑦𝐴}
Distinct variable groups:   𝑥,𝑦   𝑦,𝐴
Allowed substitution hint:   𝐴(𝑥)

Proof of Theorem viin
StepHypRef Expression
1 df-iin 4789 . 2 𝑥 ∈ V 𝐴 = {𝑦 ∣ ∀𝑥 ∈ V 𝑦𝐴}
2 ralv 3434 . . 3 (∀𝑥 ∈ V 𝑦𝐴 ↔ ∀𝑥 𝑦𝐴)
32abbii 2838 . 2 {𝑦 ∣ ∀𝑥 ∈ V 𝑦𝐴} = {𝑦 ∣ ∀𝑥 𝑦𝐴}
41, 3eqtri 2796 1 𝑥 ∈ V 𝐴 = {𝑦 ∣ ∀𝑥 𝑦𝐴}
Colors of variables: wff setvar class
Syntax hints:  wal 1505   = wceq 1507  wcel 2050  {cab 2752  wral 3082  Vcvv 3409   ciin 4787
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-ext 2744
This theorem depends on definitions:  df-bi 199  df-an 388  df-ex 1743  df-sb 2016  df-clab 2753  df-cleq 2765  df-clel 2840  df-ral 3087  df-v 3411  df-iin 4789
This theorem is referenced by: (None)
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