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Theorem rabiun 36080
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 4963 . . . . . 6 (𝑥 𝑦𝐴 𝐵 ↔ ∃𝑦𝐴 𝑥𝐵)
21anbi1i 625 . . . . 5 ((𝑥 𝑦𝐴 𝐵𝜑) ↔ (∃𝑦𝐴 𝑥𝐵𝜑))
3 r19.41v 3186 . . . . 5 (∃𝑦𝐴 (𝑥𝐵𝜑) ↔ (∃𝑦𝐴 𝑥𝐵𝜑))
42, 3bitr4i 278 . . . 4 ((𝑥 𝑦𝐴 𝐵𝜑) ↔ ∃𝑦𝐴 (𝑥𝐵𝜑))
54abbii 2807 . . 3 {𝑥 ∣ (𝑥 𝑦𝐴 𝐵𝜑)} = {𝑥 ∣ ∃𝑦𝐴 (𝑥𝐵𝜑)}
6 df-rab 3411 . . 3 {𝑥 𝑦𝐴 𝐵𝜑} = {𝑥 ∣ (𝑥 𝑦𝐴 𝐵𝜑)}
7 iunab 5016 . . 3 𝑦𝐴 {𝑥 ∣ (𝑥𝐵𝜑)} = {𝑥 ∣ ∃𝑦𝐴 (𝑥𝐵𝜑)}
85, 6, 73eqtr4i 2775 . 2 {𝑥 𝑦𝐴 𝐵𝜑} = 𝑦𝐴 {𝑥 ∣ (𝑥𝐵𝜑)}
9 df-rab 3411 . . . 4 {𝑥𝐵𝜑} = {𝑥 ∣ (𝑥𝐵𝜑)}
109a1i 11 . . 3 (𝑦𝐴 → {𝑥𝐵𝜑} = {𝑥 ∣ (𝑥𝐵𝜑)})
1110iuneq2i 4980 . 2 𝑦𝐴 {𝑥𝐵𝜑} = 𝑦𝐴 {𝑥 ∣ (𝑥𝐵𝜑)}
128, 11eqtr4i 2768 1 {𝑥 𝑦𝐴 𝐵𝜑} = 𝑦𝐴 {𝑥𝐵𝜑}
Colors of variables: wff setvar class
Syntax hints:  wa 397   = wceq 1542  wcel 2107  {cab 2714  wrex 3074  {crab 3410   ciun 4959
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-in 3922  df-ss 3932  df-iun 4961
This theorem is referenced by:  itg2addnclem2  36159
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