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Theorem rabiun 33738
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 4680 . . . . . 6 (𝑥 𝑦𝐴 𝐵 ↔ ∃𝑦𝐴 𝑥𝐵)
21anbi1i 617 . . . . 5 ((𝑥 𝑦𝐴 𝐵𝜑) ↔ (∃𝑦𝐴 𝑥𝐵𝜑))
3 r19.41v 3236 . . . . 5 (∃𝑦𝐴 (𝑥𝐵𝜑) ↔ (∃𝑦𝐴 𝑥𝐵𝜑))
42, 3bitr4i 269 . . . 4 ((𝑥 𝑦𝐴 𝐵𝜑) ↔ ∃𝑦𝐴 (𝑥𝐵𝜑))
54abbii 2882 . . 3 {𝑥 ∣ (𝑥 𝑦𝐴 𝐵𝜑)} = {𝑥 ∣ ∃𝑦𝐴 (𝑥𝐵𝜑)}
6 df-rab 3064 . . 3 {𝑥 𝑦𝐴 𝐵𝜑} = {𝑥 ∣ (𝑥 𝑦𝐴 𝐵𝜑)}
7 iunab 4722 . . 3 𝑦𝐴 {𝑥 ∣ (𝑥𝐵𝜑)} = {𝑥 ∣ ∃𝑦𝐴 (𝑥𝐵𝜑)}
85, 6, 73eqtr4i 2797 . 2 {𝑥 𝑦𝐴 𝐵𝜑} = 𝑦𝐴 {𝑥 ∣ (𝑥𝐵𝜑)}
9 df-rab 3064 . . . 4 {𝑥𝐵𝜑} = {𝑥 ∣ (𝑥𝐵𝜑)}
109a1i 11 . . 3 (𝑦𝐴 → {𝑥𝐵𝜑} = {𝑥 ∣ (𝑥𝐵𝜑)})
1110iuneq2i 4695 . 2 𝑦𝐴 {𝑥𝐵𝜑} = 𝑦𝐴 {𝑥 ∣ (𝑥𝐵𝜑)}
128, 11eqtr4i 2790 1 {𝑥 𝑦𝐴 𝐵𝜑} = 𝑦𝐴 {𝑥𝐵𝜑}
Colors of variables: wff setvar class
Syntax hints:  wa 384   = wceq 1652  wcel 2155  {cab 2751  wrex 3056  {crab 3059   ciun 4676
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-in 3739  df-ss 3746  df-iun 4678
This theorem is referenced by:  itg2addnclem2  33817
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