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Theorem eluni2f 40909
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 1842 . 2 (∃𝑥(𝐴𝑥𝑥𝐵) ↔ ∃𝑥(𝑥𝐵𝐴𝑥))
2 eluni2f.1 . . 3 𝑥𝐴
3 eluni2f.2 . . 3 𝑥𝐵
42, 3elunif 40812 . 2 (𝐴 𝐵 ↔ ∃𝑥(𝐴𝑥𝑥𝐵))
5 df-rex 3111 . 2 (∃𝑥𝐵 𝐴𝑥 ↔ ∃𝑥(𝑥𝐵𝐴𝑥))
61, 4, 53bitr4i 304 1 (𝐴 𝐵 ↔ ∃𝑥𝐵 𝐴𝑥)
Colors of variables: wff setvar class
Syntax hints:  wb 207  wa 396  wex 1761  wcel 2081  wnfc 2933  wrex 3106   cuni 4745
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-ext 2769
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-rex 3111  df-v 3439  df-uni 4746
This theorem is referenced by:  smfresal  42605  smfpimbor1lem2  42616
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