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Theorem clel3g 3661
Description: Alternate definition of membership in a set. (Contributed by NM, 13-Aug-2005.)
Assertion
Ref Expression
clel3g (𝐵𝑉 → (𝐴𝐵 ↔ ∃𝑥(𝑥 = 𝐵𝐴𝑥)))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem clel3g
StepHypRef Expression
1 eleq2 2828 . . 3 (𝑥 = 𝐵 → (𝐴𝑥𝐴𝐵))
21ceqsexgv 3654 . 2 (𝐵𝑉 → (∃𝑥(𝑥 = 𝐵𝐴𝑥) ↔ 𝐴𝐵))
32bicomd 223 1 (𝐵𝑉 → (𝐴𝐵 ↔ ∃𝑥(𝑥 = 𝐵𝐴𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wex 1776  wcel 2106
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814
This theorem is referenced by:  clel3  3662  uniprg  4928  dfiun2gOLD  5036
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