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Theorem rabiun 35346
 Description: Abstraction restricted to an indexed union. (Contributed by Brendan Leahy, 26-Oct-2017.)
Assertion
Ref Expression
rabiun {𝑥 𝑦𝐴 𝐵𝜑} = 𝑦𝐴 {𝑥𝐵𝜑}
Distinct variable groups:   𝜑,𝑦   𝑥,𝐴   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥)   𝐴(𝑦)   𝐵(𝑥,𝑦)

Proof of Theorem rabiun
StepHypRef Expression
1 eliun 4891 . . . . . 6 (𝑥 𝑦𝐴 𝐵 ↔ ∃𝑦𝐴 𝑥𝐵)
21anbi1i 626 . . . . 5 ((𝑥 𝑦𝐴 𝐵𝜑) ↔ (∃𝑦𝐴 𝑥𝐵𝜑))
3 r19.41v 3266 . . . . 5 (∃𝑦𝐴 (𝑥𝐵𝜑) ↔ (∃𝑦𝐴 𝑥𝐵𝜑))
42, 3bitr4i 281 . . . 4 ((𝑥 𝑦𝐴 𝐵𝜑) ↔ ∃𝑦𝐴 (𝑥𝐵𝜑))
54abbii 2824 . . 3 {𝑥 ∣ (𝑥 𝑦𝐴 𝐵𝜑)} = {𝑥 ∣ ∃𝑦𝐴 (𝑥𝐵𝜑)}
6 df-rab 3080 . . 3 {𝑥 𝑦𝐴 𝐵𝜑} = {𝑥 ∣ (𝑥 𝑦𝐴 𝐵𝜑)}
7 iunab 4944 . . 3 𝑦𝐴 {𝑥 ∣ (𝑥𝐵𝜑)} = {𝑥 ∣ ∃𝑦𝐴 (𝑥𝐵𝜑)}
85, 6, 73eqtr4i 2792 . 2 {𝑥 𝑦𝐴 𝐵𝜑} = 𝑦𝐴 {𝑥 ∣ (𝑥𝐵𝜑)}
9 df-rab 3080 . . . 4 {𝑥𝐵𝜑} = {𝑥 ∣ (𝑥𝐵𝜑)}
109a1i 11 . . 3 (𝑦𝐴 → {𝑥𝐵𝜑} = {𝑥 ∣ (𝑥𝐵𝜑)})
1110iuneq2i 4908 . 2 𝑦𝐴 {𝑥𝐵𝜑} = 𝑦𝐴 {𝑥 ∣ (𝑥𝐵𝜑)}
128, 11eqtr4i 2785 1 {𝑥 𝑦𝐴 𝐵𝜑} = 𝑦𝐴 {𝑥𝐵𝜑}
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 399   = wceq 1539   ∈ wcel 2112  {cab 2736  ∃wrex 3072  {crab 3075  ∪ ciun 4887 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2071  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ral 3076  df-rex 3077  df-rab 3080  df-v 3412  df-in 3868  df-ss 3878  df-iun 4889 This theorem is referenced by:  itg2addnclem2  35425
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