Theorem bj-nnfnt 37225

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 37227). Intuitionistically, ⊢ (Ⅎ'𝑥¬ 𝜑 ↔ Ⅎ'𝑥¬ ¬ 𝜑). See nfnt 1876. (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 1802	. . 3 ⊢ ((∃𝑥𝜑 → 𝜑) ↔ (¬ 𝜑 → ∀𝑥 ¬ 𝜑))
2		alimex 1851	. . 3 ⊢ ((𝜑 → ∀𝑥𝜑) ↔ (∃𝑥 ¬ 𝜑 → ¬ 𝜑))
3	1, 2	anbi12ci 638	. 2 ⊢ (((∃𝑥𝜑 → 𝜑) ∧ (𝜑 → ∀𝑥𝜑)) ↔ ((∃𝑥 ¬ 𝜑 → ¬ 𝜑) ∧ (¬ 𝜑 → ∀𝑥 ¬ 𝜑)))
4		df-bj-nnf 37202	. 2 ⊢ (Ⅎ'𝑥𝜑 ↔ ((∃𝑥𝜑 → 𝜑) ∧ (𝜑 → ∀𝑥𝜑)))
5		df-bj-nnf 37202	. 2 ⊢ (Ⅎ'𝑥 ¬ 𝜑 ↔ ((∃𝑥 ¬ 𝜑 → ¬ 𝜑) ∧ (¬ 𝜑 → ∀𝑥 ¬ 𝜑)))
6	3, 4, 5	3bitr4i 305	1 ⊢ (Ⅎ'𝑥𝜑 ↔ Ⅎ'𝑥 ¬ 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 399  ∀wal 1558  ∃wex 1799  Ⅎ'wnnf 37201

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829

This theorem depends on definitions:  df-bi 209  df-an 400  df-ex 1800  df-bj-nnf 37202

This theorem is referenced by:  bj-nnfnth  37226  bj-equsexvwd  37248