Theorem bj-dfnnf3 36759

Description: Alternate definition of nonfreeness when sp 2182 is available. (Contributed by BJ, 28-Jul-2023.) The proof should not rely on df-nf 1783. (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-dfnnf3	⊢ (Ⅎ'𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))

Proof of Theorem bj-dfnnf3

Step	Hyp	Ref	Expression
1		bj-nnfea 36736	. 2 ⊢ (Ⅎ'𝑥𝜑 → (∃𝑥𝜑 → ∀𝑥𝜑))
2		bj-19.21bit 36692	. . 3 ⊢ ((∃𝑥𝜑 → ∀𝑥𝜑) → (∃𝑥𝜑 → 𝜑))
3		bj-19.23bit 36693	. . 3 ⊢ ((∃𝑥𝜑 → ∀𝑥𝜑) → (𝜑 → ∀𝑥𝜑))
4		df-bj-nnf 36726	. . 3 ⊢ (Ⅎ'𝑥𝜑 ↔ ((∃𝑥𝜑 → 𝜑) ∧ (𝜑 → ∀𝑥𝜑)))
5	2, 3, 4	sylanbrc 583	. 2 ⊢ ((∃𝑥𝜑 → ∀𝑥𝜑) → Ⅎ'𝑥𝜑)
6	1, 5	impbii 209	1 ⊢ (Ⅎ'𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))

Syntax hints:   → wi 4   ↔ wb 206  ∀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  ax-5 1909  ax-6 1966  ax-7 2006  ax-12 2176

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1779  df-bj-nnf 36726

This theorem is referenced by:  bj-nfnnfTEMP  36760