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Theorem ssrabdv 3345
 Description: Subclass of a restricted class abstraction (deduction rule). (Contributed by NM, 31-Aug-2006.)
Hypotheses
Ref Expression
ssrabdv.1 (φB A)
ssrabdv.2 ((φ x B) → ψ)
Assertion
Ref Expression
ssrabdv (φB {x A ψ})
Distinct variable groups:   x,A   x,B   φ,x
Allowed substitution hint:   ψ(x)

Proof of Theorem ssrabdv
StepHypRef Expression
1 ssrabdv.1 . 2 (φB A)
2 ssrabdv.2 . . 3 ((φ x B) → ψ)
32ralrimiva 2697 . 2 (φx B ψ)
4 ssrab 3344 . 2 (B {x A ψ} ↔ (B A x B ψ))
51, 3, 4sylanbrc 645 1 (φB {x A ψ})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358   ∈ wcel 1710  ∀wral 2614  {crab 2618   ⊆ 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-rab 2623  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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