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Theorem elunirab 4817
 Description: Membership in union of a class abstraction. (Contributed by NM, 4-Oct-2006.)
Assertion
Ref Expression
elunirab (𝐴 {𝑥𝐵𝜑} ↔ ∃𝑥𝐵 (𝐴𝑥𝜑))
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)

Proof of Theorem elunirab
StepHypRef Expression
1 eluniab 4816 . 2 (𝐴 {𝑥 ∣ (𝑥𝐵𝜑)} ↔ ∃𝑥(𝐴𝑥 ∧ (𝑥𝐵𝜑)))
2 df-rab 3115 . . . 4 {𝑥𝐵𝜑} = {𝑥 ∣ (𝑥𝐵𝜑)}
32unieqi 4814 . . 3 {𝑥𝐵𝜑} = {𝑥 ∣ (𝑥𝐵𝜑)}
43eleq2i 2881 . 2 (𝐴 {𝑥𝐵𝜑} ↔ 𝐴 {𝑥 ∣ (𝑥𝐵𝜑)})
5 df-rex 3112 . . 3 (∃𝑥𝐵 (𝐴𝑥𝜑) ↔ ∃𝑥(𝑥𝐵 ∧ (𝐴𝑥𝜑)))
6 an12 644 . . . 4 ((𝑥𝐵 ∧ (𝐴𝑥𝜑)) ↔ (𝐴𝑥 ∧ (𝑥𝐵𝜑)))
76exbii 1849 . . 3 (∃𝑥(𝑥𝐵 ∧ (𝐴𝑥𝜑)) ↔ ∃𝑥(𝐴𝑥 ∧ (𝑥𝐵𝜑)))
85, 7bitri 278 . 2 (∃𝑥𝐵 (𝐴𝑥𝜑) ↔ ∃𝑥(𝐴𝑥 ∧ (𝑥𝐵𝜑)))
91, 4, 83bitr4i 306 1 (𝐴 {𝑥𝐵𝜑} ↔ ∃𝑥𝐵 (𝐴𝑥𝜑))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   ∧ wa 399  ∃wex 1781   ∈ wcel 2111  {cab 2776  ∃wrex 3107  {crab 3110  ∪ cuni 4801 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-rex 3112  df-rab 3115  df-v 3443  df-in 3888  df-ss 3898  df-uni 4802 This theorem is referenced by:  neiptopuni  21749  cmpcov2  22009  tgcmp  22020  hauscmplem  22025  conncompid  22050  alexsubALT  22670  cvmliftlem15  32694  fnessref  33854  cover2  35192
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