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Theorem risset 2402
Description: Two ways to say "𝐴 belongs to 𝐵." (Contributed by NM, 22-Nov-1994.)
Assertion
Ref Expression
risset (𝐴𝐵 ↔ ∃𝑥𝐵 𝑥 = 𝐴)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem risset
StepHypRef Expression
1 exancom 1542 . 2 (∃𝑥(𝑥𝐵𝑥 = 𝐴) ↔ ∃𝑥(𝑥 = 𝐴𝑥𝐵))
2 df-rex 2361 . 2 (∃𝑥𝐵 𝑥 = 𝐴 ↔ ∃𝑥(𝑥𝐵𝑥 = 𝐴))
3 df-clel 2081 . 2 (𝐴𝐵 ↔ ∃𝑥(𝑥 = 𝐴𝑥𝐵))
41, 2, 33bitr4ri 211 1 (𝐴𝐵 ↔ ∃𝑥𝐵 𝑥 = 𝐴)
Colors of variables: wff set class
Syntax hints:  wa 102  wb 103   = wceq 1287  wex 1424  wcel 1436  wrex 2356
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1379  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-4 1443  ax-ial 1470
This theorem depends on definitions:  df-bi 115  df-clel 2081  df-rex 2361
This theorem is referenced by:  reueq  2803  reuind  2809  0el  3294  iunid  3770  sucel  4213  reusv3  4258  fvmptt  5359  releldm2  5914  qsid  6311  rerecclap  8139  nndiv  8400  zq  9046  4fvwrd4  9482  bj-bdcel  11216
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