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Theorem iinab 4613
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 2793 . . . 4 𝑦𝐴
2 nfab1 2795 . . . 4 𝑦{𝑦𝜑}
31, 2nfiin 4581 . . 3 𝑦 𝑥𝐴 {𝑦𝜑}
4 nfab1 2795 . . 3 𝑦{𝑦 ∣ ∀𝑥𝐴 𝜑}
53, 4cleqf 2819 . 2 ( 𝑥𝐴 {𝑦𝜑} = {𝑦 ∣ ∀𝑥𝐴 𝜑} ↔ ∀𝑦(𝑦 𝑥𝐴 {𝑦𝜑} ↔ 𝑦 ∈ {𝑦 ∣ ∀𝑥𝐴 𝜑}))
6 abid 2639 . . . 4 (𝑦 ∈ {𝑦𝜑} ↔ 𝜑)
76ralbii 3009 . . 3 (∀𝑥𝐴 𝑦 ∈ {𝑦𝜑} ↔ ∀𝑥𝐴 𝜑)
8 vex 3234 . . . 4 𝑦 ∈ V
9 eliin 4557 . . . 4 (𝑦 ∈ V → (𝑦 𝑥𝐴 {𝑦𝜑} ↔ ∀𝑥𝐴 𝑦 ∈ {𝑦𝜑}))
108, 9ax-mp 5 . . 3 (𝑦 𝑥𝐴 {𝑦𝜑} ↔ ∀𝑥𝐴 𝑦 ∈ {𝑦𝜑})
11 abid 2639 . . 3 (𝑦 ∈ {𝑦 ∣ ∀𝑥𝐴 𝜑} ↔ ∀𝑥𝐴 𝜑)
127, 10, 113bitr4i 292 . 2 (𝑦 𝑥𝐴 {𝑦𝜑} ↔ 𝑦 ∈ {𝑦 ∣ ∀𝑥𝐴 𝜑})
135, 12mpgbir 1766 1 𝑥𝐴 {𝑦𝜑} = {𝑦 ∣ ∀𝑥𝐴 𝜑}
Colors of variables: wff setvar class
Syntax hints:  wb 196   = wceq 1523  wcel 2030  {cab 2637  wral 2941  Vcvv 3231   ciin 4553
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-v 3233  df-iin 4555
This theorem is referenced by:  iinrab  4614
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