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Theorem eluni2f 39112
 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 1786 . 2 (∃𝑥(𝐴𝑥𝑥𝐵) ↔ ∃𝑥(𝑥𝐵𝐴𝑥))
2 eluni2f.1 . . 3 𝑥𝐴
3 eluni2f.2 . . 3 𝑥𝐵
42, 3elunif 39001 . 2 (𝐴 𝐵 ↔ ∃𝑥(𝐴𝑥𝑥𝐵))
5 df-rex 2917 . 2 (∃𝑥𝐵 𝐴𝑥 ↔ ∃𝑥(𝑥𝐵𝐴𝑥))
61, 4, 53bitr4i 292 1 (𝐴 𝐵 ↔ ∃𝑥𝐵 𝐴𝑥)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 196   ∧ wa 384  ∃wex 1703   ∈ wcel 1989  Ⅎwnfc 2750  ∃wrex 2912  ∪ cuni 4434 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-rex 2917  df-v 3200  df-uni 4435 This theorem is referenced by:  smfresal  40764  smfpimbor1lem2  40775
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