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Theorem rabn0m 3390
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 2422 . 2 (∃𝑥𝐴 𝜑 ↔ ∃𝑥(𝑥𝐴𝜑))
2 rabid 2606 . . 3 (𝑥 ∈ {𝑥𝐴𝜑} ↔ (𝑥𝐴𝜑))
32exbii 1584 . 2 (∃𝑥 𝑥 ∈ {𝑥𝐴𝜑} ↔ ∃𝑥(𝑥𝐴𝜑))
4 nfv 1508 . . 3 𝑦 𝑥 ∈ {𝑥𝐴𝜑}
5 df-rab 2425 . . . . 5 {𝑥𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}
65eleq2i 2206 . . . 4 (𝑦 ∈ {𝑥𝐴𝜑} ↔ 𝑦 ∈ {𝑥 ∣ (𝑥𝐴𝜑)})
7 nfsab1 2129 . . . 4 𝑥 𝑦 ∈ {𝑥 ∣ (𝑥𝐴𝜑)}
86, 7nfxfr 1450 . . 3 𝑥 𝑦 ∈ {𝑥𝐴𝜑}
9 eleq1 2202 . . 3 (𝑥 = 𝑦 → (𝑥 ∈ {𝑥𝐴𝜑} ↔ 𝑦 ∈ {𝑥𝐴𝜑}))
104, 8, 9cbvex 1729 . 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 1468  wcel 1480  {cab 2125  wrex 2417  {crab 2420
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 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-rex 2422  df-rab 2425
This theorem is referenced by:  exss  4149  cc4f  7084  cc4n  7086
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