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Theorem nf2 1701
Description: Alternate definition of non-freeness. (Contributed by BJ, 16-Sep-2021.)
Assertion
Ref Expression
nf2 (Ⅎ𝑥𝜑 ↔ (∀𝑥𝜑 ∨ ¬ ∃𝑥𝜑))

Proof of Theorem nf2
StepHypRef Expression
1 df-nf 1700 . 2 (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
2 imor 426 . 2 ((∃𝑥𝜑 → ∀𝑥𝜑) ↔ (¬ ∃𝑥𝜑 ∨ ∀𝑥𝜑))
3 orcom 400 . 2 ((¬ ∃𝑥𝜑 ∨ ∀𝑥𝜑) ↔ (∀𝑥𝜑 ∨ ¬ ∃𝑥𝜑))
41, 2, 33bitri 284 1 (Ⅎ𝑥𝜑 ↔ (∀𝑥𝜑 ∨ ¬ ∃𝑥𝜑))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wo 381  wal 1472  wex 1694  wnf 1698
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8
This theorem depends on definitions:  df-bi 195  df-or 383  df-nf 1700
This theorem is referenced by:  nf3  1702  nfntht  1709
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