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Theorem nf4 1868
 Description: Variable x is effectively not free in φ iff φ is always true or always false. (Contributed by Mario Carneiro, 24-Sep-2016.)
Assertion
Ref Expression
nf4 (Ⅎxφ ↔ (xφ x ¬ φ))

Proof of Theorem nf4
StepHypRef Expression
1 nf2 1866 . 2 (Ⅎxφ ↔ (xφxφ))
2 imor 401 . 2 ((xφxφ) ↔ (¬ xφ xφ))
3 orcom 376 . . 3 ((¬ xφ xφ) ↔ (xφ ¬ xφ))
4 alnex 1543 . . . 4 (x ¬ φ ↔ ¬ xφ)
54orbi2i 505 . . 3 ((xφ x ¬ φ) ↔ (xφ ¬ xφ))
63, 5bitr4i 243 . 2 ((¬ xφ xφ) ↔ (xφ x ¬ φ))
71, 2, 63bitri 262 1 (Ⅎxφ ↔ (xφ x ¬ φ))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∨ wo 357  ∀wal 1540  ∃wex 1541  Ⅎwnf 1544 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-11 1746 This theorem depends on definitions:  df-bi 177  df-or 359  df-ex 1542  df-nf 1545 This theorem is referenced by: (None)
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