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Theorem eluni2f 39772
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 1947 . 2 (∃𝑥(𝐴𝑥𝑥𝐵) ↔ ∃𝑥(𝑥𝐵𝐴𝑥))
2 eluni2f.1 . . 3 𝑥𝐴
3 eluni2f.2 . . 3 𝑥𝐵
42, 3elunif 39663 . 2 (𝐴 𝐵 ↔ ∃𝑥(𝐴𝑥𝑥𝐵))
5 df-rex 3098 . 2 (∃𝑥𝐵 𝐴𝑥 ↔ ∃𝑥(𝑥𝐵𝐴𝑥))
61, 4, 53bitr4i 294 1 (𝐴 𝐵 ↔ ∃𝑥𝐵 𝐴𝑥)
Colors of variables: wff setvar class
Syntax hints:  wb 197  wa 384  wex 1859  wcel 2155  wnfc 2931  wrex 3093   cuni 4623
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2067  ax-7 2103  ax-9 2164  ax-10 2184  ax-11 2200  ax-12 2213  ax-13 2419  ax-ext 2781
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2060  df-clab 2789  df-cleq 2795  df-clel 2798  df-nfc 2933  df-rex 3098  df-v 3389  df-uni 4624
This theorem is referenced by:  smfresal  41471  smfpimbor1lem2  41482
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