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Theorem clel2g 3600
 Description: An alternate definition of class membership when the class is a set. (Contributed by NM, 18-Aug-1993.) (Revised by BJ, 12-Feb-2022.)
Assertion
Ref Expression
clel2g (𝐴𝑉 → (𝐴𝐵 ↔ ∀𝑥(𝑥 = 𝐴𝑥𝐵)))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem clel2g
StepHypRef Expression
1 nfv 1915 . . 3 𝑥 𝐴𝐵
2 eleq1 2877 . . 3 (𝑥 = 𝐴 → (𝑥𝐵𝐴𝐵))
31, 2ceqsalg 3476 . 2 (𝐴𝑉 → (∀𝑥(𝑥 = 𝐴𝑥𝐵) ↔ 𝐴𝐵))
43bicomd 226 1 (𝐴𝑉 → (𝐴𝐵 ↔ ∀𝑥(𝑥 = 𝐴𝑥𝐵)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1536   = wceq 1538   ∈ wcel 2111 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-12 2175  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-3an 1086  df-ex 1782  df-nf 1786  df-cleq 2791  df-clel 2870 This theorem is referenced by:  clel2  3601  snssg  4678
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