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Theorem nf4dc 1649
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 1650, holds for all propositions. (Contributed by Jim Kingdon, 21-Jul-2018.)
Assertion
Ref Expression
nf4dc (DECID𝑥𝜑 → (Ⅎ𝑥𝜑 ↔ (∀𝑥𝜑 ∨ ∀𝑥 ¬ 𝜑)))

Proof of Theorem nf4dc
StepHypRef Expression
1 nf2 1647 . . 3 (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
2 imordc 883 . . 3 (DECID𝑥𝜑 → ((∃𝑥𝜑 → ∀𝑥𝜑) ↔ (¬ ∃𝑥𝜑 ∨ ∀𝑥𝜑)))
31, 2syl5bb 191 . 2 (DECID𝑥𝜑 → (Ⅎ𝑥𝜑 ↔ (¬ ∃𝑥𝜑 ∨ ∀𝑥𝜑)))
4 orcom 718 . . 3 ((¬ ∃𝑥𝜑 ∨ ∀𝑥𝜑) ↔ (∀𝑥𝜑 ∨ ¬ ∃𝑥𝜑))
5 alnex 1476 . . . 4 (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)
65orbi2i 752 . . 3 ((∀𝑥𝜑 ∨ ∀𝑥 ¬ 𝜑) ↔ (∀𝑥𝜑 ∨ ¬ ∃𝑥𝜑))
74, 6bitr4i 186 . 2 ((¬ ∃𝑥𝜑 ∨ ∀𝑥𝜑) ↔ (∀𝑥𝜑 ∨ ∀𝑥 ¬ 𝜑))
83, 7syl6bb 195 1 (DECID𝑥𝜑 → (Ⅎ𝑥𝜑 ↔ (∀𝑥𝜑 ∨ ∀𝑥 ¬ 𝜑)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 104  wo 698  DECID wdc 820  wal 1330  wnf 1437  wex 1469
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-gen 1426  ax-ie2 1471  ax-4 1488  ax-ial 1515
This theorem depends on definitions:  df-bi 116  df-dc 821  df-tru 1335  df-fal 1338  df-nf 1438
This theorem is referenced by: (None)
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