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Theorem ssrabdv 3013
Description: Subclass of a restricted class abstraction (deduction rule). (Contributed by NM, 31-Aug-2006.)
Hypotheses
Ref Expression
ssrabdv.1 (φBA)
ssrabdv.2 ((φ x B) → ψ)
Assertion
Ref Expression
ssrabdv (φB ⊆ {x Aψ})
Distinct variable groups:   x,A   x,B   φ,x
Allowed substitution hint:   ψ(x)

Proof of Theorem ssrabdv
StepHypRef Expression
1 ssrabdv.1 . 2 (φBA)
2 ssrabdv.2 . . 3 ((φ x B) → ψ)
32ralrimiva 2386 . 2 (φx B ψ)
4 ssrab 3012 . 2 (B ⊆ {x Aψ} ↔ (BA x B ψ))
51, 3, 4sylanbrc 394 1 (φB ⊆ {x Aψ})
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   wcel 1390  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: (None)
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