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Theorem ssab 3004
 Description: Subclass of a class abstraction. (Contributed by NM, 16-Aug-2006.)
Assertion
Ref Expression
ssab (A ⊆ {xφ} ↔ x(x Aφ))
Distinct variable group:   x,A
Allowed substitution hint:   φ(x)

Proof of Theorem ssab
StepHypRef Expression
1 abid2 2155 . . 3 {xx A} = A
21sseq1i 2963 . 2 ({xx A} ⊆ {xφ} ↔ A ⊆ {xφ})
3 ss2ab 3002 . 2 ({xx A} ⊆ {xφ} ↔ x(x Aφ))
42, 3bitr3i 175 1 (A ⊆ {xφ} ↔ x(x Aφ))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98  ∀wal 1240   ∈ wcel 1390  {cab 2023   ⊆ 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-in 2918  df-ss 2925 This theorem is referenced by:  ssabral  3005  ssrab  3012
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