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Theorem iinab 4517
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 2751 . . . 4 𝑦𝐴
2 nfab1 2753 . . . 4 𝑦{𝑦𝜑}
31, 2nfiin 4485 . . 3 𝑦 𝑥𝐴 {𝑦𝜑}
4 nfab1 2753 . . 3 𝑦{𝑦 ∣ ∀𝑥𝐴 𝜑}
53, 4cleqf 2776 . 2 ( 𝑥𝐴 {𝑦𝜑} = {𝑦 ∣ ∀𝑥𝐴 𝜑} ↔ ∀𝑦(𝑦 𝑥𝐴 {𝑦𝜑} ↔ 𝑦 ∈ {𝑦 ∣ ∀𝑥𝐴 𝜑}))
6 abid 2598 . . . 4 (𝑦 ∈ {𝑦𝜑} ↔ 𝜑)
76ralbii 2963 . . 3 (∀𝑥𝐴 𝑦 ∈ {𝑦𝜑} ↔ ∀𝑥𝐴 𝜑)
8 vex 3176 . . . 4 𝑦 ∈ V
9 eliin 4461 . . . 4 (𝑦 ∈ V → (𝑦 𝑥𝐴 {𝑦𝜑} ↔ ∀𝑥𝐴 𝑦 ∈ {𝑦𝜑}))
108, 9ax-mp 5 . . 3 (𝑦 𝑥𝐴 {𝑦𝜑} ↔ ∀𝑥𝐴 𝑦 ∈ {𝑦𝜑})
11 abid 2598 . . 3 (𝑦 ∈ {𝑦 ∣ ∀𝑥𝐴 𝜑} ↔ ∀𝑥𝐴 𝜑)
127, 10, 113bitr4i 291 . 2 (𝑦 𝑥𝐴 {𝑦𝜑} ↔ 𝑦 ∈ {𝑦 ∣ ∀𝑥𝐴 𝜑})
135, 12mpgbir 1717 1 𝑥𝐴 {𝑦𝜑} = {𝑦 ∣ ∀𝑥𝐴 𝜑}
Colors of variables: wff setvar class
Syntax hints:  wb 195   = wceq 1475  wcel 1977  {cab 2596  wral 2896  Vcvv 3173   ciin 4456
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-v 3175  df-iin 4458
This theorem is referenced by:  iinrab  4518
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