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Theorem iinab 4508
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 2747 . . . 4 𝑦𝐴
2 nfab1 2749 . . . 4 𝑦{𝑦𝜑}
31, 2nfiin 4476 . . 3 𝑦 𝑥𝐴 {𝑦𝜑}
4 nfab1 2749 . . 3 𝑦{𝑦 ∣ ∀𝑥𝐴 𝜑}
53, 4cleqf 2772 . 2 ( 𝑥𝐴 {𝑦𝜑} = {𝑦 ∣ ∀𝑥𝐴 𝜑} ↔ ∀𝑦(𝑦 𝑥𝐴 {𝑦𝜑} ↔ 𝑦 ∈ {𝑦 ∣ ∀𝑥𝐴 𝜑}))
6 abid 2594 . . . 4 (𝑦 ∈ {𝑦𝜑} ↔ 𝜑)
76ralbii 2959 . . 3 (∀𝑥𝐴 𝑦 ∈ {𝑦𝜑} ↔ ∀𝑥𝐴 𝜑)
8 vex 3172 . . . 4 𝑦 ∈ V
9 eliin 4452 . . . 4 (𝑦 ∈ V → (𝑦 𝑥𝐴 {𝑦𝜑} ↔ ∀𝑥𝐴 𝑦 ∈ {𝑦𝜑}))
108, 9ax-mp 5 . . 3 (𝑦 𝑥𝐴 {𝑦𝜑} ↔ ∀𝑥𝐴 𝑦 ∈ {𝑦𝜑})
11 abid 2594 . . 3 (𝑦 ∈ {𝑦 ∣ ∀𝑥𝐴 𝜑} ↔ ∀𝑥𝐴 𝜑)
127, 10, 113bitr4i 290 . 2 (𝑦 𝑥𝐴 {𝑦𝜑} ↔ 𝑦 ∈ {𝑦 ∣ ∀𝑥𝐴 𝜑})
135, 12mpgbir 1716 1 𝑥𝐴 {𝑦𝜑} = {𝑦 ∣ ∀𝑥𝐴 𝜑}
Colors of variables: wff setvar class
Syntax hints:  wb 194   = wceq 1474  wcel 1976  {cab 2592  wral 2892  Vcvv 3169   ciin 4447
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2229  ax-ext 2586
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2593  df-cleq 2599  df-clel 2602  df-nfc 2736  df-ral 2897  df-v 3171  df-iin 4449
This theorem is referenced by:  iinrab  4509
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