Theorem bj-nnfa 34910

Description: Nonfreeness implies the equivalent of ax-5 1913. See nf5r 2187. (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 34906	. 2 ⊢ (Ⅎ'𝑥𝜑 ↔ ((∃𝑥𝜑 → 𝜑) ∧ (𝜑 → ∀𝑥𝜑)))
2	1	simprbi 497	1 ⊢ (Ⅎ'𝑥𝜑 → (𝜑 → ∀𝑥𝜑))

Syntax hints:   → wi 4  ∀wal 1537  ∃wex 1782  Ⅎ'wnnf 34905

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 206  df-an 397  df-bj-nnf 34906

This theorem is referenced by:  bj-nnfad  34911  bj-nnfai  34912  bj-nnfea  34916  bj-nnfim1  34926  bj-nnfim2  34927  bj-nnf-alrim  34937  bj-19.23t  34952  bj-19.37im  34954  bj-19.42t  34955  bj-sbft  34957