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Theorem iinab 3844
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 2258 . . . 4 𝑦𝐴
2 nfab1 2260 . . . 4 𝑦{𝑦𝜑}
31, 2nfiinxy 3810 . . 3 𝑦 𝑥𝐴 {𝑦𝜑}
4 nfab1 2260 . . 3 𝑦{𝑦 ∣ ∀𝑥𝐴 𝜑}
53, 4cleqf 2282 . 2 ( 𝑥𝐴 {𝑦𝜑} = {𝑦 ∣ ∀𝑥𝐴 𝜑} ↔ ∀𝑦(𝑦 𝑥𝐴 {𝑦𝜑} ↔ 𝑦 ∈ {𝑦 ∣ ∀𝑥𝐴 𝜑}))
6 abid 2105 . . . 4 (𝑦 ∈ {𝑦𝜑} ↔ 𝜑)
76ralbii 2418 . . 3 (∀𝑥𝐴 𝑦 ∈ {𝑦𝜑} ↔ ∀𝑥𝐴 𝜑)
8 vex 2663 . . . 4 𝑦 ∈ V
9 eliin 3788 . . . 4 (𝑦 ∈ V → (𝑦 𝑥𝐴 {𝑦𝜑} ↔ ∀𝑥𝐴 𝑦 ∈ {𝑦𝜑}))
108, 9ax-mp 5 . . 3 (𝑦 𝑥𝐴 {𝑦𝜑} ↔ ∀𝑥𝐴 𝑦 ∈ {𝑦𝜑})
11 abid 2105 . . 3 (𝑦 ∈ {𝑦 ∣ ∀𝑥𝐴 𝜑} ↔ ∀𝑥𝐴 𝜑)
127, 10, 113bitr4i 211 . 2 (𝑦 𝑥𝐴 {𝑦𝜑} ↔ 𝑦 ∈ {𝑦 ∣ ∀𝑥𝐴 𝜑})
135, 12mpgbir 1414 1 𝑥𝐴 {𝑦𝜑} = {𝑦 ∣ ∀𝑥𝐴 𝜑}
Colors of variables: wff set class
Syntax hints:  wb 104   = wceq 1316  wcel 1465  {cab 2103  wral 2393  Vcvv 2660   ciin 3784
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-v 2662  df-iin 3786
This theorem is referenced by:  iinrabm  3845
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