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Theorem rabiun 33695
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 4676 . . . . . 6 (𝑥 𝑦𝐴 𝐵 ↔ ∃𝑦𝐴 𝑥𝐵)
21anbi1i 733 . . . . 5 ((𝑥 𝑦𝐴 𝐵𝜑) ↔ (∃𝑦𝐴 𝑥𝐵𝜑))
3 r19.41v 3227 . . . . 5 (∃𝑦𝐴 (𝑥𝐵𝜑) ↔ (∃𝑦𝐴 𝑥𝐵𝜑))
42, 3bitr4i 267 . . . 4 ((𝑥 𝑦𝐴 𝐵𝜑) ↔ ∃𝑦𝐴 (𝑥𝐵𝜑))
54abbii 2877 . . 3 {𝑥 ∣ (𝑥 𝑦𝐴 𝐵𝜑)} = {𝑥 ∣ ∃𝑦𝐴 (𝑥𝐵𝜑)}
6 df-rab 3059 . . 3 {𝑥 𝑦𝐴 𝐵𝜑} = {𝑥 ∣ (𝑥 𝑦𝐴 𝐵𝜑)}
7 iunab 4718 . . 3 𝑦𝐴 {𝑥 ∣ (𝑥𝐵𝜑)} = {𝑥 ∣ ∃𝑦𝐴 (𝑥𝐵𝜑)}
85, 6, 73eqtr4i 2792 . 2 {𝑥 𝑦𝐴 𝐵𝜑} = 𝑦𝐴 {𝑥 ∣ (𝑥𝐵𝜑)}
9 df-rab 3059 . . . 4 {𝑥𝐵𝜑} = {𝑥 ∣ (𝑥𝐵𝜑)}
109a1i 11 . . 3 (𝑦𝐴 → {𝑥𝐵𝜑} = {𝑥 ∣ (𝑥𝐵𝜑)})
1110iuneq2i 4691 . 2 𝑦𝐴 {𝑥𝐵𝜑} = 𝑦𝐴 {𝑥 ∣ (𝑥𝐵𝜑)}
128, 11eqtr4i 2785 1 {𝑥 𝑦𝐴 𝐵𝜑} = 𝑦𝐴 {𝑥𝐵𝜑}
Colors of variables: wff setvar class
Syntax hints:  wa 383   = wceq 1632  wcel 2139  {cab 2746  wrex 3051  {crab 3054   ciun 4672
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-in 3722  df-ss 3729  df-iun 4674
This theorem is referenced by:  itg2addnclem2  33775
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