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Theorem nf4dc 1557
 Description: Variable x is effectively not free in φ iff φ is always true or always false, given a decidability condition. The reverse direction, nf4r 1558, holds for all propositions. (Contributed by Jim Kingdon, 21-Jul-2018.)
Assertion
Ref Expression
nf4dc (DECID xφ → (Ⅎxφ ↔ (xφ x ¬ φ)))

Proof of Theorem nf4dc
StepHypRef Expression
1 nf2 1555 . . 3 (Ⅎxφ ↔ (xφxφ))
2 imordc 795 . . 3 (DECID xφ → ((xφxφ) ↔ (¬ xφ xφ)))
31, 2syl5bb 181 . 2 (DECID xφ → (Ⅎxφ ↔ (¬ xφ xφ)))
4 orcom 646 . . 3 ((¬ xφ xφ) ↔ (xφ ¬ xφ))
5 alnex 1385 . . . 4 (x ¬ φ ↔ ¬ xφ)
65orbi2i 678 . . 3 ((xφ x ¬ φ) ↔ (xφ ¬ xφ))
74, 6bitr4i 176 . 2 ((¬ xφ xφ) ↔ (xφ x ¬ φ))
83, 7syl6bb 185 1 (DECID xφ → (Ⅎxφ ↔ (xφ x ¬ φ)))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 98   ∨ wo 628  DECID wdc 741  ∀wal 1240  Ⅎwnf 1346  ∃wex 1378 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-gen 1335  ax-ie2 1380  ax-4 1397  ax-ial 1424 This theorem depends on definitions:  df-bi 110  df-dc 742  df-tru 1245  df-fal 1248  df-nf 1347 This theorem is referenced by: (None)
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