Theorem bj-nnfa 37151

Description: Nonfreeness implies the equivalent of ax-5 1924. See nf5r 2223. (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 37150	. 2 ⊢ (Ⅎ'𝑥𝜑 ↔ ((∃𝑥𝜑 → 𝜑) ∧ (𝜑 → ∀𝑥𝜑)))
2	1	simprbi 500	1 ⊢ (Ⅎ'𝑥𝜑 → (𝜑 → ∀𝑥𝜑))

Syntax hints:   → wi 4  ∀wal 1552  ∃wex 1793  Ⅎ'wnnf 37149

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 209  df-an 399  df-bj-nnf 37150

This theorem is referenced by:  bj-nnfad  37152  bj-nnfai  37153  bj-nnfea  37157  bj-nnfim1  37164  bj-nnfim2  37165  bj-nnf-alrim  37168  bj-19.23t  37185  bj-19.37im  37187  bj-19.42t  37188  bj-sbft  37201