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Theorem clel2 3327
 Description: An alternate definition of class membership when the class is a set. (Contributed by NM, 18-Aug-1993.)
Hypothesis
Ref Expression
clel2.1 𝐴 ∈ V
Assertion
Ref Expression
clel2 (𝐴𝐵 ↔ ∀𝑥(𝑥 = 𝐴𝑥𝐵))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem clel2
StepHypRef Expression
1 clel2.1 . . 3 𝐴 ∈ V
2 eleq1 2686 . . 3 (𝑥 = 𝐴 → (𝑥𝐵𝐴𝐵))
31, 2ceqsalv 3223 . 2 (∀𝑥(𝑥 = 𝐴𝑥𝐵) ↔ 𝐴𝐵)
43bicomi 214 1 (𝐴𝐵 ↔ ∀𝑥(𝑥 = 𝐴𝑥𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196  ∀wal 1478   = wceq 1480   ∈ wcel 1987  Vcvv 3190 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-12 2044  ax-ext 2601 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-v 3192 This theorem is referenced by:  snss  4293  mptelee  25709
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