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Theorem risset 2518
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 1619 . 2 (∃𝑥(𝑥𝐵𝑥 = 𝐴) ↔ ∃𝑥(𝑥 = 𝐴𝑥𝐵))
2 df-rex 2474 . 2 (∃𝑥𝐵 𝑥 = 𝐴 ↔ ∃𝑥(𝑥𝐵𝑥 = 𝐴))
3 df-clel 2185 . 2 (𝐴𝐵 ↔ ∃𝑥(𝑥 = 𝐴𝑥𝐵))
41, 2, 33bitr4ri 213 1 (𝐴𝐵 ↔ ∃𝑥𝐵 𝑥 = 𝐴)
Colors of variables: wff set class
Syntax hints:  wa 104  wb 105   = wceq 1364  wex 1503  wcel 2160  wrex 2469
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-4 1521  ax-ial 1545
This theorem depends on definitions:  df-bi 117  df-clel 2185  df-rex 2474
This theorem is referenced by:  clel5  2889  reueq  2951  reuind  2957  0el  3460  iunid  3957  sucel  4428  reusv3  4478  fvmptt  5627  releldm2  6209  qsid  6625  rerecclap  8716  nndiv  8989  zq  9655  4fvwrd4  10169  conjnmzb  13216  bj-bdcel  15042
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