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Theorem ssabral 3337
 Description: The relation for a subclass of a class abstraction is equivalent to restricted quantification. (Contributed by NM, 6-Sep-2006.)
Assertion
Ref Expression
ssabral (A {x φ} ↔ x A φ)
Distinct variable group:   x,A
Allowed substitution hint:   φ(x)

Proof of Theorem ssabral
StepHypRef Expression
1 ssab 3336 . 2 (A {x φ} ↔ x(x Aφ))
2 df-ral 2619 . 2 (x A φx(x Aφ))
31, 2bitr4i 243 1 (A {x φ} ↔ x A φ)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540   ∈ wcel 1710  {cab 2339  ∀wral 2614   ⊆ wss 3257 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ral 2619  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259 This theorem is referenced by: (None)
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