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Theorem bj-nnfnt 36742
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 36748). Intuitionistically, (Ⅎ'𝑥¬ 𝜑 ↔ Ⅎ'𝑥¬ ¬ 𝜑). See nfnt 1855. (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 1781 . . 3 ((∃𝑥𝜑𝜑) ↔ (¬ 𝜑 → ∀𝑥 ¬ 𝜑))
2 alimex 1830 . . 3 ((𝜑 → ∀𝑥𝜑) ↔ (∃𝑥 ¬ 𝜑 → ¬ 𝜑))
31, 2anbi12ci 629 . 2 (((∃𝑥𝜑𝜑) ∧ (𝜑 → ∀𝑥𝜑)) ↔ ((∃𝑥 ¬ 𝜑 → ¬ 𝜑) ∧ (¬ 𝜑 → ∀𝑥 ¬ 𝜑)))
4 df-bj-nnf 36726 . 2 (Ⅎ'𝑥𝜑 ↔ ((∃𝑥𝜑𝜑) ∧ (𝜑 → ∀𝑥𝜑)))
5 df-bj-nnf 36726 . 2 (Ⅎ'𝑥 ¬ 𝜑 ↔ ((∃𝑥 ¬ 𝜑 → ¬ 𝜑) ∧ (¬ 𝜑 → ∀𝑥 ¬ 𝜑)))
63, 4, 53bitr4i 303 1 (Ⅎ'𝑥𝜑 ↔ Ⅎ'𝑥 ¬ 𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wal 1537  wex 1778  Ⅎ'wnnf 36725
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1779  df-bj-nnf 36726
This theorem is referenced by:  bj-nnfnth  36745  bj-equsexvwd  36783
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