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Theorem eluni2f 42147
 Description: Membership in class union. Restricted quantifier version. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
Hypotheses
Ref Expression
eluni2f.1 𝑥𝐴
eluni2f.2 𝑥𝐵
Assertion
Ref Expression
eluni2f (𝐴 𝐵 ↔ ∃𝑥𝐵 𝐴𝑥)
Distinct variable group:   𝐴,𝐵
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)

Proof of Theorem eluni2f
StepHypRef Expression
1 exancom 1862 . 2 (∃𝑥(𝐴𝑥𝑥𝐵) ↔ ∃𝑥(𝑥𝐵𝐴𝑥))
2 eluni2f.1 . . 3 𝑥𝐴
3 eluni2f.2 . . 3 𝑥𝐵
42, 3elunif 42053 . 2 (𝐴 𝐵 ↔ ∃𝑥(𝐴𝑥𝑥𝐵))
5 df-rex 3076 . 2 (∃𝑥𝐵 𝐴𝑥 ↔ ∃𝑥(𝑥𝐵𝐴𝑥))
61, 4, 53bitr4i 306 1 (𝐴 𝐵 ↔ ∃𝑥𝐵 𝐴𝑥)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   ∧ wa 399  ∃wex 1781   ∈ wcel 2111  Ⅎwnfc 2899  ∃wrex 3071  ∪ 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 2729 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 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-rex 3076  df-v 3411  df-uni 4802 This theorem is referenced by:  smfresal  43821  smfpimbor1lem2  43832
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