Theorem bj-nfnnfTEMP 34108

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 1784 except via df-nf 1784 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 34107	. 2 ⊢ (Ⅎ'𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
2		df-nf 1784	. 2 ⊢ (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
3	1, 2	bitr4i 280	1 ⊢ (Ⅎ'𝑥𝜑 ↔ Ⅎ𝑥𝜑)

Syntax hints:   → wi 4   ↔ wb 208  ∀wal 1534  ∃wex 1779  Ⅎwnf 1783  Ⅎ'wnnf 34076

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-12 2176

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1780  df-nf 1784  df-bj-nnf 34077

This theorem is referenced by: (None)