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Theorem iunab 3806
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 2240 . . . 4 𝑦𝐴
2 nfab1 2242 . . . 4 𝑦{𝑦𝜑}
31, 2nfiunxy 3786 . . 3 𝑦 𝑥𝐴 {𝑦𝜑}
4 nfab1 2242 . . 3 𝑦{𝑦 ∣ ∃𝑥𝐴 𝜑}
53, 4cleqf 2264 . 2 ( 𝑥𝐴 {𝑦𝜑} = {𝑦 ∣ ∃𝑥𝐴 𝜑} ↔ ∀𝑦(𝑦 𝑥𝐴 {𝑦𝜑} ↔ 𝑦 ∈ {𝑦 ∣ ∃𝑥𝐴 𝜑}))
6 abid 2088 . . . 4 (𝑦 ∈ {𝑦𝜑} ↔ 𝜑)
76rexbii 2401 . . 3 (∃𝑥𝐴 𝑦 ∈ {𝑦𝜑} ↔ ∃𝑥𝐴 𝜑)
8 eliun 3764 . . 3 (𝑦 𝑥𝐴 {𝑦𝜑} ↔ ∃𝑥𝐴 𝑦 ∈ {𝑦𝜑})
9 abid 2088 . . 3 (𝑦 ∈ {𝑦 ∣ ∃𝑥𝐴 𝜑} ↔ ∃𝑥𝐴 𝜑)
107, 8, 93bitr4i 211 . 2 (𝑦 𝑥𝐴 {𝑦𝜑} ↔ 𝑦 ∈ {𝑦 ∣ ∃𝑥𝐴 𝜑})
115, 10mpgbir 1397 1 𝑥𝐴 {𝑦𝜑} = {𝑦 ∣ ∃𝑥𝐴 𝜑}
Colors of variables: wff set class
Syntax hints:  wb 104   = wceq 1299  wcel 1448  {cab 2086  wrex 2376   ciun 3760
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082
This theorem depends on definitions:  df-bi 116  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-rex 2381  df-v 2643  df-iun 3762
This theorem is referenced by:  iunrab  3807  iunid  3815  dfimafn2  5403
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