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Theorem rabss 3011
Description: Restricted class abstraction in a subclass relationship. (Contributed by NM, 16-Aug-2006.)
Assertion
Ref Expression
rabss ({x Aφ} ⊆ Bx A (φx B))
Distinct variable group:   x,B
Allowed substitution hints:   φ(x)   A(x)

Proof of Theorem rabss
StepHypRef Expression
1 df-rab 2309 . . 3 {x Aφ} = {x ∣ (x A φ)}
21sseq1i 2963 . 2 ({x Aφ} ⊆ B ↔ {x ∣ (x A φ)} ⊆ B)
3 abss 3003 . 2 ({x ∣ (x A φ)} ⊆ Bx((x A φ) → x B))
4 impexp 250 . . . 4 (((x A φ) → x B) ↔ (x A → (φx B)))
54albii 1356 . . 3 (x((x A φ) → x B) ↔ x(x A → (φx B)))
6 df-ral 2305 . . 3 (x A (φx B) ↔ x(x A → (φx B)))
75, 6bitr4i 176 . 2 (x((x A φ) → x B) ↔ x A (φx B))
82, 3, 73bitri 195 1 ({x Aφ} ⊆ Bx A (φx B))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98  wal 1240   wcel 1390  {cab 2023  wral 2300  {crab 2304  wss 2911
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-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  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-nfc 2164  df-ral 2305  df-rab 2309  df-in 2918  df-ss 2925
This theorem is referenced by:  rabssdv  3014
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