Theorem bj-nfnnfTEMP 34567

Description: New nonfreeness is equivalent to old nonfreeness on core FOL axioms plus sp 2183. (Contributed by BJ, 28-Jul-2023.) The proof should not rely on df-nf 1791 except via df-nf 1791 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 34566	. 2 ⊢ (Ⅎ'𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
2		df-nf 1791	. 2 ⊢ (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
3	1, 2	bitr4i 281	1 ⊢ (Ⅎ'𝑥𝜑 ↔ Ⅎ𝑥𝜑)

Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1540  ∃wex 1786  Ⅎwnf 1790  Ⅎ'wnnf 34532

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-12 2178

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1787  df-nf 1791  df-bj-nnf 34533

This theorem is referenced by: (None)