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Theorem iunrab 3920
Description: The indexed union of a restricted class abstraction. (Contributed by NM, 3-Jan-2004.) (Proof shortened by Mario Carneiro, 14-Nov-2016.)
Assertion
Ref Expression
iunrab 𝑥𝐴 {𝑦𝐵𝜑} = {𝑦𝐵 ∣ ∃𝑥𝐴 𝜑}
Distinct variable groups:   𝑦,𝐴   𝑥,𝑦   𝑥,𝐵
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥)   𝐵(𝑦)

Proof of Theorem iunrab
StepHypRef Expression
1 iunab 3919 . 2 𝑥𝐴 {𝑦 ∣ (𝑦𝐵𝜑)} = {𝑦 ∣ ∃𝑥𝐴 (𝑦𝐵𝜑)}
2 df-rab 2457 . . . 4 {𝑦𝐵𝜑} = {𝑦 ∣ (𝑦𝐵𝜑)}
32a1i 9 . . 3 (𝑥𝐴 → {𝑦𝐵𝜑} = {𝑦 ∣ (𝑦𝐵𝜑)})
43iuneq2i 3891 . 2 𝑥𝐴 {𝑦𝐵𝜑} = 𝑥𝐴 {𝑦 ∣ (𝑦𝐵𝜑)}
5 df-rab 2457 . . 3 {𝑦𝐵 ∣ ∃𝑥𝐴 𝜑} = {𝑦 ∣ (𝑦𝐵 ∧ ∃𝑥𝐴 𝜑)}
6 r19.42v 2627 . . . 4 (∃𝑥𝐴 (𝑦𝐵𝜑) ↔ (𝑦𝐵 ∧ ∃𝑥𝐴 𝜑))
76abbii 2286 . . 3 {𝑦 ∣ ∃𝑥𝐴 (𝑦𝐵𝜑)} = {𝑦 ∣ (𝑦𝐵 ∧ ∃𝑥𝐴 𝜑)}
85, 7eqtr4i 2194 . 2 {𝑦𝐵 ∣ ∃𝑥𝐴 𝜑} = {𝑦 ∣ ∃𝑥𝐴 (𝑦𝐵𝜑)}
91, 4, 83eqtr4i 2201 1 𝑥𝐴 {𝑦𝐵𝜑} = {𝑦𝐵 ∣ ∃𝑥𝐴 𝜑}
Colors of variables: wff set class
Syntax hints:  wa 103   = wceq 1348  wcel 2141  {cab 2156  wrex 2449  {crab 2452   ciun 3873
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-in 3127  df-ss 3134  df-iun 3875
This theorem is referenced by:  hashrabrex  11444  phisum  12194
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