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Theorem iunab 3866
Description: The indexed union of a class abstraction. (Contributed by NM, 27-Dec-2004.)
Assertion
Ref Expression
iunab 𝑥𝐴 {𝑦𝜑} = {𝑦 ∣ ∃𝑥𝐴 𝜑}
Distinct variable groups:   𝑦,𝐴   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥)

Proof of Theorem iunab
StepHypRef Expression
1 nfcv 2282 . . . 4 𝑦𝐴
2 nfab1 2284 . . . 4 𝑦{𝑦𝜑}
31, 2nfiunxy 3846 . . 3 𝑦 𝑥𝐴 {𝑦𝜑}
4 nfab1 2284 . . 3 𝑦{𝑦 ∣ ∃𝑥𝐴 𝜑}
53, 4cleqf 2306 . 2 ( 𝑥𝐴 {𝑦𝜑} = {𝑦 ∣ ∃𝑥𝐴 𝜑} ↔ ∀𝑦(𝑦 𝑥𝐴 {𝑦𝜑} ↔ 𝑦 ∈ {𝑦 ∣ ∃𝑥𝐴 𝜑}))
6 abid 2128 . . . 4 (𝑦 ∈ {𝑦𝜑} ↔ 𝜑)
76rexbii 2445 . . 3 (∃𝑥𝐴 𝑦 ∈ {𝑦𝜑} ↔ ∃𝑥𝐴 𝜑)
8 eliun 3824 . . 3 (𝑦 𝑥𝐴 {𝑦𝜑} ↔ ∃𝑥𝐴 𝑦 ∈ {𝑦𝜑})
9 abid 2128 . . 3 (𝑦 ∈ {𝑦 ∣ ∃𝑥𝐴 𝜑} ↔ ∃𝑥𝐴 𝜑)
107, 8, 93bitr4i 211 . 2 (𝑦 𝑥𝐴 {𝑦𝜑} ↔ 𝑦 ∈ {𝑦 ∣ ∃𝑥𝐴 𝜑})
115, 10mpgbir 1430 1 𝑥𝐴 {𝑦𝜑} = {𝑦 ∣ ∃𝑥𝐴 𝜑}
Colors of variables: wff set class
Syntax hints:  wb 104   = wceq 1332  wcel 1481  {cab 2126  wrex 2418   ciun 3820
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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-iun 3822
This theorem is referenced by:  iunrab  3867  iunid  3875  dfimafn2  5478
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