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Theorem risset 2369
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 1515 . 2 (∃𝑥(𝑥𝐵𝑥 = 𝐴) ↔ ∃𝑥(𝑥 = 𝐴𝑥𝐵))
2 df-rex 2329 . 2 (∃𝑥𝐵 𝑥 = 𝐴 ↔ ∃𝑥(𝑥𝐵𝑥 = 𝐴))
3 df-clel 2052 . 2 (𝐴𝐵 ↔ ∃𝑥(𝑥 = 𝐴𝑥𝐵))
41, 2, 33bitr4ri 206 1 (𝐴𝐵 ↔ ∃𝑥𝐵 𝑥 = 𝐴)
Colors of variables: wff set class
Syntax hints:  wa 101  wb 102   = wceq 1259  wex 1397  wcel 1409  wrex 2324
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-4 1416  ax-ial 1443
This theorem depends on definitions:  df-bi 114  df-clel 2052  df-rex 2329
This theorem is referenced by:  reueq  2761  reuind  2767  0el  3269  iunid  3740  sucel  4175  reusv3  4220  fvmptt  5290  releldm2  5839  qsid  6202  rerecclap  7781  nndiv  8030  zq  8658  4fvwrd4  9099  bj-bdcel  10344
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