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Theorem clel2 2790
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 2178 . . 3 (𝑥 = 𝐴 → (𝑥𝐵𝐴𝐵))
31, 2ceqsalv 2688 . 2 (∀𝑥(𝑥 = 𝐴𝑥𝐵) ↔ 𝐴𝐵)
43bicomi 131 1 (𝐴𝐵 ↔ ∀𝑥(𝑥 = 𝐴𝑥𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104  wal 1312   = wceq 1314  wcel 1463  Vcvv 2658
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1406  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-v 2660
This theorem is referenced by:  snss  3617
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