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Theorem rabn0m 3222
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 2290 . 2 (x A φx(x A φ))
2 rabid 2463 . . 3 (x {x Aφ} ↔ (x A φ))
32exbii 1478 . 2 (x x {x Aφ} ↔ x(x A φ))
4 nfv 1402 . . 3 y x {x Aφ}
5 df-rab 2293 . . . . 5 {x Aφ} = {x ∣ (x A φ)}
65eleq2i 2086 . . . 4 (y {x Aφ} ↔ y {x ∣ (x A φ)})
7 nfsab1 2012 . . . 4 x y {x ∣ (x A φ)}
86, 7nfxfr 1343 . . 3 x y {x Aφ}
9 eleq1 2082 . . 3 (x = y → (x {x Aφ} ↔ y {x Aφ}))
104, 8, 9cbvex 1621 . 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 1362   wcel 1374  {cab 2008  wrex 2285  {crab 2288
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 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-11 1378  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-rex 2290  df-rab 2293
This theorem is referenced by:  exss  3937
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