Theorem bj-nfnnfTEMP 36760

Description: New nonfreeness is equivalent to old nonfreeness on core FOL axioms plus sp 2182. (Contributed by BJ, 28-Jul-2023.) The proof should not rely on df-nf 1783 except via df-nf 1783 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 36759	. 2 ⊢ (Ⅎ'𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
2		df-nf 1783	. 2 ⊢ (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
3	1, 2	bitr4i 278	1 ⊢ (Ⅎ'𝑥𝜑 ↔ Ⅎ𝑥𝜑)

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1537  ∃wex 1778  Ⅎwnf 1782  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-nf 1783  df-bj-nnf 36726

This theorem is referenced by: (None)