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Theorem dfsb3 2056
 Description: An alternate definition of proper substitution df-sb 1649 that uses only primitive connectives (no defined terms) on the right-hand side. (Contributed by NM, 6-Mar-2007.)
Assertion
Ref Expression
dfsb3 ([y / x]φ ↔ ((x = y → ¬ φ) → x(x = yφ)))

Proof of Theorem dfsb3
StepHypRef Expression
1 df-or 359 . 2 (((x = y φ) x(x = yφ)) ↔ (¬ (x = y φ) → x(x = yφ)))
2 dfsb2 2055 . 2 ([y / x]φ ↔ ((x = y φ) x(x = yφ)))
3 imnan 411 . . 3 ((x = y → ¬ φ) ↔ ¬ (x = y φ))
43imbi1i 315 . 2 (((x = y → ¬ φ) → x(x = yφ)) ↔ (¬ (x = y φ) → x(x = yφ)))
51, 2, 43bitr4i 268 1 ([y / x]φ ↔ ((x = y → ¬ φ) → x(x = yφ)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∨ wo 357   ∧ wa 358  ∀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-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649 This theorem is referenced by: (None)
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