Theorem bj-nfnnfTEMP 36741

Description: New nonfreeness is equivalent to old nonfreeness on core FOL axioms plus sp 2181. (Contributed by BJ, 28-Jul-2023.) The proof should not rely on df-nf 1781 except via df-nf 1781 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 36740	. 2 ⊢ (Ⅎ'𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
2		df-nf 1781	. 2 ⊢ (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
3	1, 2	bitr4i 278	1 ⊢ (Ⅎ'𝑥𝜑 ↔ Ⅎ𝑥𝜑)

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1535  ∃wex 1776  Ⅎwnf 1780  Ⅎ'wnnf 36706

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-12 2175

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1777  df-nf 1781  df-bj-nnf 36707

This theorem is referenced by: (None)