Theorem bj-dfnnf3 37465

Description: Alternate definition of nonfreeness when sp 2222 is available. (Contributed by BJ, 28-Jul-2023.) The proof should not rely on df-nf 1817. (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 37418	. 2 ⊢ (Ⅎ'𝑥𝜑 → (∃𝑥𝜑 → ∀𝑥𝜑))
2		bj-19.21bit 37374	. . 3 ⊢ ((∃𝑥𝜑 → ∀𝑥𝜑) → (∃𝑥𝜑 → 𝜑))
3		bj-19.23bit 37375	. . 3 ⊢ ((∃𝑥𝜑 → ∀𝑥𝜑) → (𝜑 → ∀𝑥𝜑))
4		df-bj-nnf 37411	. . 3 ⊢ (Ⅎ'𝑥𝜑 ↔ ((∃𝑥𝜑 → 𝜑) ∧ (𝜑 → ∀𝑥𝜑)))
5	2, 3, 4	sylanbrc 595	. 2 ⊢ ((∃𝑥𝜑 → ∀𝑥𝜑) → Ⅎ'𝑥𝜑)
6	1, 5	impbii 212	1 ⊢ (Ⅎ'𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  ∀wal 1568  ∃wex 1812  Ⅎ'wnnf 37410

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-12 2216

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-bj-nnf 37411

This theorem is used by:  bj-nfnnfTEMP  37466