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Theorem elunirab 4927
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 4926 . 2 (𝐴 {𝑥 ∣ (𝑥𝐵𝜑)} ↔ ∃𝑥(𝐴𝑥 ∧ (𝑥𝐵𝜑)))
2 df-rab 3434 . . . 4 {𝑥𝐵𝜑} = {𝑥 ∣ (𝑥𝐵𝜑)}
32unieqi 4924 . . 3 {𝑥𝐵𝜑} = {𝑥 ∣ (𝑥𝐵𝜑)}
43eleq2i 2831 . 2 (𝐴 {𝑥𝐵𝜑} ↔ 𝐴 {𝑥 ∣ (𝑥𝐵𝜑)})
5 df-rex 3069 . . 3 (∃𝑥𝐵 (𝐴𝑥𝜑) ↔ ∃𝑥(𝑥𝐵 ∧ (𝐴𝑥𝜑)))
6 an12 645 . . . 4 ((𝑥𝐵 ∧ (𝐴𝑥𝜑)) ↔ (𝐴𝑥 ∧ (𝑥𝐵𝜑)))
76exbii 1845 . . 3 (∃𝑥(𝑥𝐵 ∧ (𝐴𝑥𝜑)) ↔ ∃𝑥(𝐴𝑥 ∧ (𝑥𝐵𝜑)))
85, 7bitri 275 . 2 (∃𝑥𝐵 (𝐴𝑥𝜑) ↔ ∃𝑥(𝐴𝑥 ∧ (𝑥𝐵𝜑)))
91, 4, 83bitr4i 303 1 (𝐴 {𝑥𝐵𝜑} ↔ ∃𝑥𝐵 (𝐴𝑥𝜑))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395  wex 1776  wcel 2106  {cab 2712  wrex 3068  {crab 3433   cuni 4912
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rex 3069  df-rab 3434  df-v 3480  df-ss 3980  df-uni 4913
This theorem is referenced by:  neiptopuni  23154  cmpcov2  23414  tgcmp  23425  hauscmplem  23430  conncompid  23455  alexsubALT  24075  cvmliftlem15  35283  fnessref  36340  cover2  37702
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