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Theorem bj-nnfnt 34549
Description: A variable is nonfree in a formula if and only if it is nonfree in its negation. The foward implication is intuitionistically valid (and that direction is sufficient for the purpose of recursively proving that some formulas have a given variable not free in them, like bj-nnfim 34555). Intuitionistically, (Ⅎ'𝑥¬ 𝜑 ↔ Ⅎ'𝑥¬ ¬ 𝜑). See nfnt 1862. (Contributed by BJ, 28-Jul-2023.)
Assertion
Ref Expression
bj-nnfnt (Ⅎ'𝑥𝜑 ↔ Ⅎ'𝑥 ¬ 𝜑)

Proof of Theorem bj-nnfnt
StepHypRef Expression
1 eximal 1789 . . 3 ((∃𝑥𝜑𝜑) ↔ (¬ 𝜑 → ∀𝑥 ¬ 𝜑))
2 alimex 1837 . . 3 ((𝜑 → ∀𝑥𝜑) ↔ (∃𝑥 ¬ 𝜑 → ¬ 𝜑))
31, 2anbi12ci 631 . 2 (((∃𝑥𝜑𝜑) ∧ (𝜑 → ∀𝑥𝜑)) ↔ ((∃𝑥 ¬ 𝜑 → ¬ 𝜑) ∧ (¬ 𝜑 → ∀𝑥 ¬ 𝜑)))
4 df-bj-nnf 34533 . 2 (Ⅎ'𝑥𝜑 ↔ ((∃𝑥𝜑𝜑) ∧ (𝜑 → ∀𝑥𝜑)))
5 df-bj-nnf 34533 . 2 (Ⅎ'𝑥 ¬ 𝜑 ↔ ((∃𝑥 ¬ 𝜑 → ¬ 𝜑) ∧ (¬ 𝜑 → ∀𝑥 ¬ 𝜑)))
63, 4, 53bitr4i 306 1 (Ⅎ'𝑥𝜑 ↔ Ⅎ'𝑥 ¬ 𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  wal 1540  wex 1786  Ⅎ'wnnf 34532
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1787  df-bj-nnf 34533
This theorem is referenced by:  bj-nnfnth  34552
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