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Theorem rabn0m 3242
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 2309 . 2 (∃𝑥𝐴 𝜑 ↔ ∃𝑥(𝑥𝐴𝜑))
2 rabid 2482 . . 3 (𝑥 ∈ {𝑥𝐴𝜑} ↔ (𝑥𝐴𝜑))
32exbii 1496 . 2 (∃𝑥 𝑥 ∈ {𝑥𝐴𝜑} ↔ ∃𝑥(𝑥𝐴𝜑))
4 nfv 1421 . . 3 𝑦 𝑥 ∈ {𝑥𝐴𝜑}
5 df-rab 2312 . . . . 5 {𝑥𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}
65eleq2i 2104 . . . 4 (𝑦 ∈ {𝑥𝐴𝜑} ↔ 𝑦 ∈ {𝑥 ∣ (𝑥𝐴𝜑)})
7 nfsab1 2030 . . . 4 𝑥 𝑦 ∈ {𝑥 ∣ (𝑥𝐴𝜑)}
86, 7nfxfr 1363 . . 3 𝑥 𝑦 ∈ {𝑥𝐴𝜑}
9 eleq1 2100 . . 3 (𝑥 = 𝑦 → (𝑥 ∈ {𝑥𝐴𝜑} ↔ 𝑦 ∈ {𝑥𝐴𝜑}))
104, 8, 9cbvex 1639 . 2 (∃𝑥 𝑥 ∈ {𝑥𝐴𝜑} ↔ ∃𝑦 𝑦 ∈ {𝑥𝐴𝜑})
111, 3, 103bitr2ri 198 1 (∃𝑦 𝑦 ∈ {𝑥𝐴𝜑} ↔ ∃𝑥𝐴 𝜑)
Colors of variables: wff set class
Syntax hints:  wa 97  wb 98  wex 1381  wcel 1393  {cab 2026  wrex 2304  {crab 2307
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-rex 2309  df-rab 2312
This theorem is referenced by:  exss  3959
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