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Theorem rabbidva 2850
 Description: Equivalent wff's yield equal restricted class abstractions (deduction rule). (Contributed by NM, 28-Nov-2003.)
Hypothesis
Ref Expression
rabbidva.1 ((φ x A) → (ψχ))
Assertion
Ref Expression
rabbidva (φ → {x A ψ} = {x A χ})
Distinct variable group:   φ,x
Allowed substitution hints:   ψ(x)   χ(x)   A(x)

Proof of Theorem rabbidva
StepHypRef Expression
1 rabbidva.1 . . 3 ((φ x A) → (ψχ))
21ralrimiva 2697 . 2 (φx A (ψχ))
3 rabbi 2789 . 2 (x A (ψχ) ↔ {x A ψ} = {x A χ})
42, 3sylib 188 1 (φ → {x A ψ} = {x A χ})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ∀wral 2614  {crab 2618 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-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-ral 2619  df-rab 2623 This theorem is referenced by:  rabbidv  2851  rabeqbidva  2855  rabbi2dva  3463
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