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Theorem rabn0m 3239
 Description: Inhabited restricted class abstraction. (Contributed by Jim Kingdon, 18-Sep-2018.)
Assertion
Ref Expression
rabn0m (y y {x Aφ} ↔ x A 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 2306 . 2 (x A φx(x A φ))
2 rabid 2479 . . 3 (x {x Aφ} ↔ (x A φ))
32exbii 1493 . 2 (x x {x Aφ} ↔ x(x A φ))
4 nfv 1418 . . 3 y x {x Aφ}
5 df-rab 2309 . . . . 5 {x Aφ} = {x ∣ (x A φ)}
65eleq2i 2101 . . . 4 (y {x Aφ} ↔ y {x ∣ (x A φ)})
7 nfsab1 2027 . . . 4 x y {x ∣ (x A φ)}
86, 7nfxfr 1360 . . 3 x y {x Aφ}
9 eleq1 2097 . . 3 (x = y → (x {x Aφ} ↔ y {x Aφ}))
104, 8, 9cbvex 1636 . 2 (x x {x Aφ} ↔ y y {x Aφ})
111, 3, 103bitr2ri 198 1 (y y {x Aφ} ↔ x A φ)
 Colors of variables: wff set class Syntax hints:   ∧ wa 97   ↔ wb 98  ∃wex 1378   ∈ wcel 1390  {cab 2023  ∃wrex 2301  {crab 2304 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 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-11 1394  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-rex 2306  df-rab 2309 This theorem is referenced by:  exss  3954
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