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Theorem ss2rabdv 3347
 Description: Deduction of restricted abstraction subclass from implication. (Contributed by NM, 30-May-2006.)
Hypothesis
Ref Expression
ss2rabdv.1 ((φ x A) → (ψχ))
Assertion
Ref Expression
ss2rabdv (φ → {x A ψ} {x A χ})
Distinct variable group:   φ,x
Allowed substitution hints:   ψ(x)   χ(x)   A(x)

Proof of Theorem ss2rabdv
StepHypRef Expression
1 ss2rabdv.1 . . 3 ((φ x A) → (ψχ))
21ralrimiva 2697 . 2 (φx A (ψχ))
3 ss2rab 3342 . 2 ({x A ψ} {x A χ} ↔ x A (ψχ))
42, 3sylibr 203 1 (φ → {x A ψ} {x A χ})
 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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