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Theorem abn0m 3291
Description: Inhabited class abstraction. (Contributed by Jim Kingdon, 8-Jul-2022.)
Assertion
Ref Expression
abn0m (∃𝑦 𝑦 ∈ {𝑥𝜑} ↔ ∃𝑥𝜑)
Distinct variable groups:   𝜑,𝑦   𝑥,𝑦
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem abn0m
StepHypRef Expression
1 nfv 1462 . . 3 𝑦 𝑥 ∈ {𝑥𝜑}
2 nfsab1 2073 . . 3 𝑥 𝑦 ∈ {𝑥𝜑}
3 eleq1w 2143 . . 3 (𝑥 = 𝑦 → (𝑥 ∈ {𝑥𝜑} ↔ 𝑦 ∈ {𝑥𝜑}))
41, 2, 3cbvex 1681 . 2 (∃𝑥 𝑥 ∈ {𝑥𝜑} ↔ ∃𝑦 𝑦 ∈ {𝑥𝜑})
5 abid 2071 . . 3 (𝑥 ∈ {𝑥𝜑} ↔ 𝜑)
65exbii 1537 . 2 (∃𝑥 𝑥 ∈ {𝑥𝜑} ↔ ∃𝑥𝜑)
74, 6bitr3i 184 1 (∃𝑦 𝑦 ∈ {𝑥𝜑} ↔ ∃𝑥𝜑)
Colors of variables: wff set class
Syntax hints:  wb 103  wex 1422  wcel 1434  {cab 2069
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 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-11 1438  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1688  df-clab 2070  df-clel 2079
This theorem is referenced by:  mapprc  6339
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