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Theorem rexun 3944
Description: Restricted existential quantification over union. (Contributed by Jeff Madsen, 5-Jan-2011.)
Assertion
Ref Expression
rexun (∃𝑥 ∈ (𝐴𝐵)𝜑 ↔ (∃𝑥𝐴 𝜑 ∨ ∃𝑥𝐵 𝜑))

Proof of Theorem rexun
StepHypRef Expression
1 df-rex 3067 . 2 (∃𝑥 ∈ (𝐴𝐵)𝜑 ↔ ∃𝑥(𝑥 ∈ (𝐴𝐵) ∧ 𝜑))
2 19.43 1962 . . 3 (∃𝑥((𝑥𝐴𝜑) ∨ (𝑥𝐵𝜑)) ↔ (∃𝑥(𝑥𝐴𝜑) ∨ ∃𝑥(𝑥𝐵𝜑)))
3 elun 3904 . . . . . 6 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
43anbi1i 610 . . . . 5 ((𝑥 ∈ (𝐴𝐵) ∧ 𝜑) ↔ ((𝑥𝐴𝑥𝐵) ∧ 𝜑))
5 andir 993 . . . . 5 (((𝑥𝐴𝑥𝐵) ∧ 𝜑) ↔ ((𝑥𝐴𝜑) ∨ (𝑥𝐵𝜑)))
64, 5bitri 264 . . . 4 ((𝑥 ∈ (𝐴𝐵) ∧ 𝜑) ↔ ((𝑥𝐴𝜑) ∨ (𝑥𝐵𝜑)))
76exbii 1924 . . 3 (∃𝑥(𝑥 ∈ (𝐴𝐵) ∧ 𝜑) ↔ ∃𝑥((𝑥𝐴𝜑) ∨ (𝑥𝐵𝜑)))
8 df-rex 3067 . . . 4 (∃𝑥𝐴 𝜑 ↔ ∃𝑥(𝑥𝐴𝜑))
9 df-rex 3067 . . . 4 (∃𝑥𝐵 𝜑 ↔ ∃𝑥(𝑥𝐵𝜑))
108, 9orbi12i 900 . . 3 ((∃𝑥𝐴 𝜑 ∨ ∃𝑥𝐵 𝜑) ↔ (∃𝑥(𝑥𝐴𝜑) ∨ ∃𝑥(𝑥𝐵𝜑)))
112, 7, 103bitr4i 292 . 2 (∃𝑥(𝑥 ∈ (𝐴𝐵) ∧ 𝜑) ↔ (∃𝑥𝐴 𝜑 ∨ ∃𝑥𝐵 𝜑))
121, 11bitri 264 1 (∃𝑥 ∈ (𝐴𝐵)𝜑 ↔ (∃𝑥𝐴 𝜑 ∨ ∃𝑥𝐵 𝜑))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 382  wo 836  wex 1852  wcel 2145  wrex 3062  cun 3721
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-rex 3067  df-v 3353  df-un 3728
This theorem is referenced by:  rexprg  4372  rextpg  4374  iunxun  4739  oarec  7794  zornn0g  9527  scshwfzeqfzo  13774  rpnnen2lem12  15153  dvdsprmpweqnn  15789  vdwlem6  15890  pmatcollpw3fi1  20806  cmpfi  21425  poimirlem25  33760  unima  39859
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