Theorem bj-nnfa 37072

Description: Nonfreeness implies the equivalent of ax-5 1917. See nf5r 2206. (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 37071	. 2 ⊢ (Ⅎ'𝑥𝜑 ↔ ((∃𝑥𝜑 → 𝜑) ∧ (𝜑 → ∀𝑥𝜑)))
2	1	simprbi 498	1 ⊢ (Ⅎ'𝑥𝜑 → (𝜑 → ∀𝑥𝜑))

Syntax hints:   → wi 4  ∀wal 1545  ∃wex 1786  Ⅎ'wnnf 37070

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 208  df-an 397  df-bj-nnf 37071

This theorem is referenced by:  bj-nnfad  37073  bj-nnfai  37074  bj-nnfea  37078  bj-nnfim1  37085  bj-nnfim2  37086  bj-nnf-alrim  37089  bj-19.23t  37106  bj-19.37im  37108  bj-19.42t  37109  bj-sbft  37122