Theorem bj-nfnnfTEMP 36288

Description: New nonfreeness is equivalent to old nonfreeness on core FOL axioms plus sp 2171. (Contributed by BJ, 28-Jul-2023.) The proof should not rely on df-nf 1778 except via df-nf 1778 directly. (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-nfnnfTEMP	⊢ (Ⅎ'𝑥𝜑 ↔ Ⅎ𝑥𝜑)

Proof of Theorem bj-nfnnfTEMP

Step	Hyp	Ref	Expression
1		bj-dfnnf3 36287	. 2 ⊢ (Ⅎ'𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
2		df-nf 1778	. 2 ⊢ (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
3	1, 2	bitr4i 277	1 ⊢ (Ⅎ'𝑥𝜑 ↔ Ⅎ𝑥𝜑)

Syntax hints:   → wi 4   ↔ wb 205  ∀wal 1531  ∃wex 1773  Ⅎwnf 1777  Ⅎ'wnnf 36253

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-12 2166

This theorem depends on definitions:  df-bi 206  df-an 395  df-ex 1774  df-nf 1778  df-bj-nnf 36254

This theorem is referenced by: (None)