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Theorem clel4 3482
Description: An alternate definition of class membership when the class is a set. (Contributed by NM, 18-Aug-1993.)
Hypothesis
Ref Expression
clel4.1 𝐵 ∈ V
Assertion
Ref Expression
clel4 (𝐴𝐵 ↔ ∀𝑥(𝑥 = 𝐵𝐴𝑥))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem clel4
StepHypRef Expression
1 clel4.1 . . 3 𝐵 ∈ V
2 eleq2 2828 . . 3 (𝑥 = 𝐵 → (𝐴𝑥𝐴𝐵))
31, 2ceqsalv 3373 . 2 (∀𝑥(𝑥 = 𝐵𝐴𝑥) ↔ 𝐴𝐵)
43bicomi 214 1 (𝐴𝐵 ↔ ∀𝑥(𝑥 = 𝐵𝐴𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wal 1630   = wceq 1632  wcel 2139  Vcvv 3340
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-12 2196  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-v 3342
This theorem is referenced by:  intpr  4662
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