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Theorem nf4r 1671
Description: If 𝜑 is always true or always false, then variable 𝑥 is effectively not free in 𝜑. The converse holds given a decidability condition, as seen at nf4dc 1670. (Contributed by Jim Kingdon, 21-Jul-2018.)
Assertion
Ref Expression
nf4r ((∀𝑥𝜑 ∨ ∀𝑥 ¬ 𝜑) → Ⅎ𝑥𝜑)

Proof of Theorem nf4r
StepHypRef Expression
1 orcom 728 . . 3 ((¬ ∃𝑥𝜑 ∨ ∀𝑥𝜑) ↔ (∀𝑥𝜑 ∨ ¬ ∃𝑥𝜑))
2 alnex 1499 . . . 4 (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)
32orbi2i 762 . . 3 ((∀𝑥𝜑 ∨ ∀𝑥 ¬ 𝜑) ↔ (∀𝑥𝜑 ∨ ¬ ∃𝑥𝜑))
41, 3bitr4i 187 . 2 ((¬ ∃𝑥𝜑 ∨ ∀𝑥𝜑) ↔ (∀𝑥𝜑 ∨ ∀𝑥 ¬ 𝜑))
5 imorr 721 . . 3 ((¬ ∃𝑥𝜑 ∨ ∀𝑥𝜑) → (∃𝑥𝜑 → ∀𝑥𝜑))
6 nf2 1668 . . 3 (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
75, 6sylibr 134 . 2 ((¬ ∃𝑥𝜑 ∨ ∀𝑥𝜑) → Ⅎ𝑥𝜑)
84, 7sylbir 135 1 ((∀𝑥𝜑 ∨ ∀𝑥 ¬ 𝜑) → Ⅎ𝑥𝜑)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wo 708  wal 1351  wnf 1460  wex 1492
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-gen 1449  ax-ie2 1494  ax-4 1510  ax-ial 1534
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-fal 1359  df-nf 1461
This theorem is referenced by: (None)
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