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Theorem iinab 4981
Description: Indexed intersection of a class builder. (Contributed by NM, 6-Dec-2011.)
Assertion
Ref Expression
iinab 𝑥𝐴 {𝑦𝜑} = {𝑦 ∣ ∀𝑥𝐴 𝜑}
Distinct variable groups:   𝑦,𝐴   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥)

Proof of Theorem iinab
StepHypRef Expression
1 nfcv 2974 . . . 4 𝑦𝐴
2 nfab1 2976 . . . 4 𝑦{𝑦𝜑}
31, 2nfiin 4941 . . 3 𝑦 𝑥𝐴 {𝑦𝜑}
4 nfab1 2976 . . 3 𝑦{𝑦 ∣ ∀𝑥𝐴 𝜑}
53, 4cleqf 3007 . 2 ( 𝑥𝐴 {𝑦𝜑} = {𝑦 ∣ ∀𝑥𝐴 𝜑} ↔ ∀𝑦(𝑦 𝑥𝐴 {𝑦𝜑} ↔ 𝑦 ∈ {𝑦 ∣ ∀𝑥𝐴 𝜑}))
6 abid 2800 . . . 4 (𝑦 ∈ {𝑦𝜑} ↔ 𝜑)
76ralbii 3162 . . 3 (∀𝑥𝐴 𝑦 ∈ {𝑦𝜑} ↔ ∀𝑥𝐴 𝜑)
8 eliin 4915 . . . 4 (𝑦 ∈ V → (𝑦 𝑥𝐴 {𝑦𝜑} ↔ ∀𝑥𝐴 𝑦 ∈ {𝑦𝜑}))
98elv 3497 . . 3 (𝑦 𝑥𝐴 {𝑦𝜑} ↔ ∀𝑥𝐴 𝑦 ∈ {𝑦𝜑})
10 abid 2800 . . 3 (𝑦 ∈ {𝑦 ∣ ∀𝑥𝐴 𝜑} ↔ ∀𝑥𝐴 𝜑)
117, 9, 103bitr4i 304 . 2 (𝑦 𝑥𝐴 {𝑦𝜑} ↔ 𝑦 ∈ {𝑦 ∣ ∀𝑥𝐴 𝜑})
125, 11mpgbir 1791 1 𝑥𝐴 {𝑦𝜑} = {𝑦 ∣ ∀𝑥𝐴 𝜑}
Colors of variables: wff setvar class
Syntax hints:  wb 207   = wceq 1528  wcel 2105  {cab 2796  wral 3135  Vcvv 3492   ciin 4911
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-v 3494  df-iin 4913
This theorem is referenced by:  iinrab  4982
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