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Theorem sbhb 2107
 Description: Two ways of expressing "x is (effectively) not free in φ." (Contributed by NM, 29-May-2009.)
Assertion
Ref Expression
sbhb ((φxφ) ↔ y(φ → [y / x]φ))
Distinct variable group:   φ,y
Allowed substitution hint:   φ(x)

Proof of Theorem sbhb
StepHypRef Expression
1 nfv 1619 . . . 4 yφ
21sb8 2092 . . 3 (xφy[y / x]φ)
32imbi2i 303 . 2 ((φxφ) ↔ (φy[y / x]φ))
4 19.21v 1890 . 2 (y(φ → [y / x]φ) ↔ (φy[y / x]φ))
53, 4bitr4i 243 1 ((φxφ) ↔ y(φ → [y / x]φ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540  [wsb 1648 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 This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649 This theorem is referenced by:  sbnf2  2108
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