Theorem bj-nnfa 34079

Description: Nonfreeness implies the equivalent of ax-5 1910. See nf5r 2192, nf5ri 2194. (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 34075	. 2 ⊢ (Ⅎ'𝑥𝜑 ↔ ((∃𝑥𝜑 → 𝜑) ∧ (𝜑 → ∀𝑥𝜑)))
2	1	simprbi 499	1 ⊢ (Ⅎ'𝑥𝜑 → (𝜑 → ∀𝑥𝜑))

Syntax hints:   → wi 4  ∀wal 1534  ∃wex 1779  Ⅎ'wnnf 34074

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 34075

This theorem is referenced by:  bj-nnfad  34080  bj-nnfea  34083  bj-nnfim1  34092  bj-nnfim2  34093  bj-nnf-alrim  34103  bj-19.23t  34118  bj-19.37im  34120  bj-19.42t  34121  bj-sbft  34123