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Theorem rabn0m 3465
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 2474 . 2 (∃𝑥𝐴 𝜑 ↔ ∃𝑥(𝑥𝐴𝜑))
2 rabid 2666 . . 3 (𝑥 ∈ {𝑥𝐴𝜑} ↔ (𝑥𝐴𝜑))
32exbii 1616 . 2 (∃𝑥 𝑥 ∈ {𝑥𝐴𝜑} ↔ ∃𝑥(𝑥𝐴𝜑))
4 nfv 1539 . . 3 𝑦 𝑥 ∈ {𝑥𝐴𝜑}
5 df-rab 2477 . . . . 5 {𝑥𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}
65eleq2i 2256 . . . 4 (𝑦 ∈ {𝑥𝐴𝜑} ↔ 𝑦 ∈ {𝑥 ∣ (𝑥𝐴𝜑)})
7 nfsab1 2179 . . . 4 𝑥 𝑦 ∈ {𝑥 ∣ (𝑥𝐴𝜑)}
86, 7nfxfr 1485 . . 3 𝑥 𝑦 ∈ {𝑥𝐴𝜑}
9 eleq1 2252 . . 3 (𝑥 = 𝑦 → (𝑥 ∈ {𝑥𝐴𝜑} ↔ 𝑦 ∈ {𝑥𝐴𝜑}))
104, 8, 9cbvex 1767 . 2 (∃𝑥 𝑥 ∈ {𝑥𝐴𝜑} ↔ ∃𝑦 𝑦 ∈ {𝑥𝐴𝜑})
111, 3, 103bitr2ri 209 1 (∃𝑦 𝑦 ∈ {𝑥𝐴𝜑} ↔ ∃𝑥𝐴 𝜑)
Colors of variables: wff set class
Syntax hints:  wa 104  wb 105  wex 1503  wcel 2160  {cab 2175  wrex 2469  {crab 2472
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-rex 2474  df-rab 2477
This theorem is referenced by:  exss  4245  cc4f  7298  cc4n  7300  nnwosdc  12072  lspf  13705
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