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Theorem abn0m 3327
 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 1473 . . 3 𝑦 𝑥 ∈ {𝑥𝜑}
2 nfsab1 2085 . . 3 𝑥 𝑦 ∈ {𝑥𝜑}
3 eleq1w 2155 . . 3 (𝑥 = 𝑦 → (𝑥 ∈ {𝑥𝜑} ↔ 𝑦 ∈ {𝑥𝜑}))
41, 2, 3cbvex 1693 . 2 (∃𝑥 𝑥 ∈ {𝑥𝜑} ↔ ∃𝑦 𝑦 ∈ {𝑥𝜑})
5 abid 2083 . . 3 (𝑥 ∈ {𝑥𝜑} ↔ 𝜑)
65exbii 1548 . 2 (∃𝑥 𝑥 ∈ {𝑥𝜑} ↔ ∃𝑥𝜑)
74, 6bitr3i 185 1 (∃𝑦 𝑦 ∈ {𝑥𝜑} ↔ ∃𝑥𝜑)
 Colors of variables: wff set class Syntax hints:   ↔ wb 104  ∃wex 1433   ∈ wcel 1445  {cab 2081 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-11 1449  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479 This theorem depends on definitions:  df-bi 116  df-nf 1402  df-sb 1700  df-clab 2082  df-clel 2091 This theorem is referenced by:  mapprc  6449
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