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Theorem rabasiun 4494
Description: A class abstraction with a restricted existential quantification expressed as indexed union. (Contributed by Alexander van der Vekens, 29-Jul-2018.)
Assertion
Ref Expression
rabasiun {𝑥𝑋 ∣ ∃𝑦𝑌 𝜑} = 𝑦𝑌 {𝑥𝑋𝜑}
Distinct variable groups:   𝑥,𝑋,𝑦   𝑥,𝑌
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝑌(𝑦)

Proof of Theorem rabasiun
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 nfcv 2767 . . . . . 6 𝑧𝑋
21nfcri 2761 . . . . 5 𝑧 𝑥𝑋
3 nfv 1845 . . . . 5 𝑧𝑦𝑌 𝜑
42, 3nfan 1830 . . . 4 𝑧(𝑥𝑋 ∧ ∃𝑦𝑌 𝜑)
5 nfcv 2767 . . . . . 6 𝑥𝑋
65nfcri 2761 . . . . 5 𝑥 𝑧𝑋
7 nfcv 2767 . . . . . 6 𝑥𝑌
8 nfs1v 2441 . . . . . 6 𝑥[𝑧 / 𝑥]𝜑
97, 8nfrex 3006 . . . . 5 𝑥𝑦𝑌 [𝑧 / 𝑥]𝜑
106, 9nfan 1830 . . . 4 𝑥(𝑧𝑋 ∧ ∃𝑦𝑌 [𝑧 / 𝑥]𝜑)
11 eleq1 2692 . . . . 5 (𝑥 = 𝑧 → (𝑥𝑋𝑧𝑋))
12 sbequ12 2113 . . . . . 6 (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
1312rexbidv 3050 . . . . 5 (𝑥 = 𝑧 → (∃𝑦𝑌 𝜑 ↔ ∃𝑦𝑌 [𝑧 / 𝑥]𝜑))
1411, 13anbi12d 746 . . . 4 (𝑥 = 𝑧 → ((𝑥𝑋 ∧ ∃𝑦𝑌 𝜑) ↔ (𝑧𝑋 ∧ ∃𝑦𝑌 [𝑧 / 𝑥]𝜑)))
154, 10, 14cbvab 2749 . . 3 {𝑥 ∣ (𝑥𝑋 ∧ ∃𝑦𝑌 𝜑)} = {𝑧 ∣ (𝑧𝑋 ∧ ∃𝑦𝑌 [𝑧 / 𝑥]𝜑)}
16 nfcv 2767 . . . . . . 7 𝑥𝑧
1716, 5, 8, 12elrabf 3348 . . . . . 6 (𝑧 ∈ {𝑥𝑋𝜑} ↔ (𝑧𝑋 ∧ [𝑧 / 𝑥]𝜑))
1817rexbii 3039 . . . . 5 (∃𝑦𝑌 𝑧 ∈ {𝑥𝑋𝜑} ↔ ∃𝑦𝑌 (𝑧𝑋 ∧ [𝑧 / 𝑥]𝜑))
19 r19.42v 3089 . . . . 5 (∃𝑦𝑌 (𝑧𝑋 ∧ [𝑧 / 𝑥]𝜑) ↔ (𝑧𝑋 ∧ ∃𝑦𝑌 [𝑧 / 𝑥]𝜑))
2018, 19bitr2i 265 . . . 4 ((𝑧𝑋 ∧ ∃𝑦𝑌 [𝑧 / 𝑥]𝜑) ↔ ∃𝑦𝑌 𝑧 ∈ {𝑥𝑋𝜑})
2120abbii 2742 . . 3 {𝑧 ∣ (𝑧𝑋 ∧ ∃𝑦𝑌 [𝑧 / 𝑥]𝜑)} = {𝑧 ∣ ∃𝑦𝑌 𝑧 ∈ {𝑥𝑋𝜑}}
2215, 21eqtri 2648 . 2 {𝑥 ∣ (𝑥𝑋 ∧ ∃𝑦𝑌 𝜑)} = {𝑧 ∣ ∃𝑦𝑌 𝑧 ∈ {𝑥𝑋𝜑}}
23 df-rab 2921 . 2 {𝑥𝑋 ∣ ∃𝑦𝑌 𝜑} = {𝑥 ∣ (𝑥𝑋 ∧ ∃𝑦𝑌 𝜑)}
24 df-iun 4492 . 2 𝑦𝑌 {𝑥𝑋𝜑} = {𝑧 ∣ ∃𝑦𝑌 𝑧 ∈ {𝑥𝑋𝜑}}
2522, 23, 243eqtr4i 2658 1 {𝑥𝑋 ∣ ∃𝑦𝑌 𝜑} = 𝑦𝑌 {𝑥𝑋𝜑}
Colors of variables: wff setvar class
Syntax hints:  wa 384   = wceq 1480  [wsb 1882  wcel 1992  {cab 2612  wrex 2913  {crab 2916   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 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606
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 1883  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-iun 4492
This theorem is referenced by:  hashrabrex  14477
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