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Theorem abn0m 3419
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 1508 . . 3 𝑦 𝑥 ∈ {𝑥𝜑}
2 nfsab1 2147 . . 3 𝑥 𝑦 ∈ {𝑥𝜑}
3 eleq1w 2218 . . 3 (𝑥 = 𝑦 → (𝑥 ∈ {𝑥𝜑} ↔ 𝑦 ∈ {𝑥𝜑}))
41, 2, 3cbvex 1736 . 2 (∃𝑥 𝑥 ∈ {𝑥𝜑} ↔ ∃𝑦 𝑦 ∈ {𝑥𝜑})
5 abid 2145 . . 3 (𝑥 ∈ {𝑥𝜑} ↔ 𝜑)
65exbii 1585 . 2 (∃𝑥 𝑥 ∈ {𝑥𝜑} ↔ ∃𝑥𝜑)
74, 6bitr3i 185 1 (∃𝑦 𝑦 ∈ {𝑥𝜑} ↔ ∃𝑥𝜑)
Colors of variables: wff set class
Syntax hints:  wb 104  wex 1472  wcel 2128  {cab 2143
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-11 1486  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514
This theorem depends on definitions:  df-bi 116  df-nf 1441  df-sb 1743  df-clab 2144  df-clel 2153
This theorem is referenced by:  mapprc  6598
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