Theorem bj-nnfa 36694

Description: Nonfreeness implies the equivalent of ax-5 1909. See nf5r 2195. (Contributed by BJ, 28-Jul-2023.)

Assertion

Ref	Expression
bj-nnfa	⊢ (Ⅎ'𝑥𝜑 → (𝜑 → ∀𝑥𝜑))

Proof of Theorem bj-nnfa

Step	Hyp	Ref	Expression
1		df-bj-nnf 36690	. 2 ⊢ (Ⅎ'𝑥𝜑 ↔ ((∃𝑥𝜑 → 𝜑) ∧ (𝜑 → ∀𝑥𝜑)))
2	1	simprbi 496	1 ⊢ (Ⅎ'𝑥𝜑 → (𝜑 → ∀𝑥𝜑))

Syntax hints:   → wi 4  ∀wal 1535  ∃wex 1777  Ⅎ'wnnf 36689

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 207  df-an 396  df-bj-nnf 36690

This theorem is referenced by:  bj-nnfad  36695  bj-nnfai  36696  bj-nnfea  36700  bj-nnfim1  36710  bj-nnfim2  36711  bj-nnf-alrim  36721  bj-19.23t  36736  bj-19.37im  36738  bj-19.42t  36739  bj-sbft  36741