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Theorem bj-nnfnt 36074
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 36080). Intuitionistically, (Ⅎ'𝑥¬ 𝜑 ↔ Ⅎ'𝑥¬ ¬ 𝜑). See nfnt 1851. (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 1776 . . 3 ((∃𝑥𝜑𝜑) ↔ (¬ 𝜑 → ∀𝑥 ¬ 𝜑))
2 alimex 1825 . . 3 ((𝜑 → ∀𝑥𝜑) ↔ (∃𝑥 ¬ 𝜑 → ¬ 𝜑))
31, 2anbi12ci 627 . 2 (((∃𝑥𝜑𝜑) ∧ (𝜑 → ∀𝑥𝜑)) ↔ ((∃𝑥 ¬ 𝜑 → ¬ 𝜑) ∧ (¬ 𝜑 → ∀𝑥 ¬ 𝜑)))
4 df-bj-nnf 36058 . 2 (Ⅎ'𝑥𝜑 ↔ ((∃𝑥𝜑𝜑) ∧ (𝜑 → ∀𝑥𝜑)))
5 df-bj-nnf 36058 . 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 205  wa 395  wal 1531  wex 1773  Ⅎ'wnnf 36057
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803
This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1774  df-bj-nnf 36058
This theorem is referenced by:  bj-nnfnth  36077  bj-equsexvwd  36115
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