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Theorem rabn0m 3360
Description: Inhabited restricted class abstraction. (Contributed by Jim Kingdon, 18-Sep-2018.)
Assertion
Ref Expression
rabn0m (∃𝑦 𝑦 ∈ {𝑥𝐴𝜑} ↔ ∃𝑥𝐴 𝜑)
Distinct variable groups:   𝑥,𝑦   𝑦,𝐴   𝜑,𝑦
Allowed substitution hints:   𝜑(𝑥)   𝐴(𝑥)

Proof of Theorem rabn0m
StepHypRef Expression
1 df-rex 2399 . 2 (∃𝑥𝐴 𝜑 ↔ ∃𝑥(𝑥𝐴𝜑))
2 rabid 2583 . . 3 (𝑥 ∈ {𝑥𝐴𝜑} ↔ (𝑥𝐴𝜑))
32exbii 1569 . 2 (∃𝑥 𝑥 ∈ {𝑥𝐴𝜑} ↔ ∃𝑥(𝑥𝐴𝜑))
4 nfv 1493 . . 3 𝑦 𝑥 ∈ {𝑥𝐴𝜑}
5 df-rab 2402 . . . . 5 {𝑥𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}
65eleq2i 2184 . . . 4 (𝑦 ∈ {𝑥𝐴𝜑} ↔ 𝑦 ∈ {𝑥 ∣ (𝑥𝐴𝜑)})
7 nfsab1 2107 . . . 4 𝑥 𝑦 ∈ {𝑥 ∣ (𝑥𝐴𝜑)}
86, 7nfxfr 1435 . . 3 𝑥 𝑦 ∈ {𝑥𝐴𝜑}
9 eleq1 2180 . . 3 (𝑥 = 𝑦 → (𝑥 ∈ {𝑥𝐴𝜑} ↔ 𝑦 ∈ {𝑥𝐴𝜑}))
104, 8, 9cbvex 1714 . 2 (∃𝑥 𝑥 ∈ {𝑥𝐴𝜑} ↔ ∃𝑦 𝑦 ∈ {𝑥𝐴𝜑})
111, 3, 103bitr2ri 208 1 (∃𝑦 𝑦 ∈ {𝑥𝐴𝜑} ↔ ∃𝑥𝐴 𝜑)
Colors of variables: wff set class
Syntax hints:  wa 103  wb 104  wex 1453  wcel 1465  {cab 2103  wrex 2394  {crab 2397
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 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-11 1469  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-rex 2399  df-rab 2402
This theorem is referenced by:  exss  4119
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