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Theorem rabiun 33041
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 4495 . . . . . 6 (𝑥 𝑦𝐴 𝐵 ↔ ∃𝑦𝐴 𝑥𝐵)
21anbi1i 730 . . . . 5 ((𝑥 𝑦𝐴 𝐵𝜑) ↔ (∃𝑦𝐴 𝑥𝐵𝜑))
3 r19.41v 3082 . . . . 5 (∃𝑦𝐴 (𝑥𝐵𝜑) ↔ (∃𝑦𝐴 𝑥𝐵𝜑))
42, 3bitr4i 267 . . . 4 ((𝑥 𝑦𝐴 𝐵𝜑) ↔ ∃𝑦𝐴 (𝑥𝐵𝜑))
54abbii 2736 . . 3 {𝑥 ∣ (𝑥 𝑦𝐴 𝐵𝜑)} = {𝑥 ∣ ∃𝑦𝐴 (𝑥𝐵𝜑)}
6 df-rab 2916 . . 3 {𝑥 𝑦𝐴 𝐵𝜑} = {𝑥 ∣ (𝑥 𝑦𝐴 𝐵𝜑)}
7 iunab 4537 . . 3 𝑦𝐴 {𝑥 ∣ (𝑥𝐵𝜑)} = {𝑥 ∣ ∃𝑦𝐴 (𝑥𝐵𝜑)}
85, 6, 73eqtr4i 2653 . 2 {𝑥 𝑦𝐴 𝐵𝜑} = 𝑦𝐴 {𝑥 ∣ (𝑥𝐵𝜑)}
9 df-rab 2916 . . . 4 {𝑥𝐵𝜑} = {𝑥 ∣ (𝑥𝐵𝜑)}
109a1i 11 . . 3 (𝑦𝐴 → {𝑥𝐵𝜑} = {𝑥 ∣ (𝑥𝐵𝜑)})
1110iuneq2i 4510 . 2 𝑦𝐴 {𝑥𝐵𝜑} = 𝑦𝐴 {𝑥 ∣ (𝑥𝐵𝜑)}
128, 11eqtr4i 2646 1 {𝑥 𝑦𝐴 𝐵𝜑} = 𝑦𝐴 {𝑥𝐵𝜑}
Colors of variables: wff setvar class
Syntax hints:  wa 384   = wceq 1480  wcel 1987  {cab 2607  wrex 2908  {crab 2911   ciun 4490
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-in 3566  df-ss 3573  df-iun 4492
This theorem is referenced by:  itg2addnclem2  33121
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