Theorem bj-nnfnt 37066

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 37068). Intuitionistically, ⊢ (Ⅎ'𝑥¬ 𝜑 ↔ Ⅎ'𝑥¬ ¬ 𝜑). See nfnt 1858. (Contributed by BJ, 28-Jul-2023.)

Assertion

Ref	Expression
bj-nnfnt	⊢ (Ⅎ'𝑥𝜑 ↔ Ⅎ'𝑥 ¬ 𝜑)

Proof of Theorem bj-nnfnt

Step	Hyp	Ref	Expression
1		eximal 1784	. . 3 ⊢ ((∃𝑥𝜑 → 𝜑) ↔ (¬ 𝜑 → ∀𝑥 ¬ 𝜑))
2		alimex 1833	. . 3 ⊢ ((𝜑 → ∀𝑥𝜑) ↔ (∃𝑥 ¬ 𝜑 → ¬ 𝜑))
3	1, 2	anbi12ci 630	. 2 ⊢ (((∃𝑥𝜑 → 𝜑) ∧ (𝜑 → ∀𝑥𝜑)) ↔ ((∃𝑥 ¬ 𝜑 → ¬ 𝜑) ∧ (¬ 𝜑 → ∀𝑥 ¬ 𝜑)))
4		df-bj-nnf 37043	. 2 ⊢ (Ⅎ'𝑥𝜑 ↔ ((∃𝑥𝜑 → 𝜑) ∧ (𝜑 → ∀𝑥𝜑)))
5		df-bj-nnf 37043	. 2 ⊢ (Ⅎ'𝑥 ¬ 𝜑 ↔ ((∃𝑥 ¬ 𝜑 → ¬ 𝜑) ∧ (¬ 𝜑 → ∀𝑥 ¬ 𝜑)))
6	3, 4, 5	3bitr4i 303	1 ⊢ (Ⅎ'𝑥𝜑 ↔ Ⅎ'𝑥 ¬ 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1540  ∃wex 1781  Ⅎ'wnnf 37042

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1782  df-bj-nnf 37043

This theorem is referenced by:  bj-nnfnth  37067  bj-equsexvwd  37089