Theorem bj-nnfa 34175

Description: Nonfreeness implies the equivalent of ax-5 1911. See nf5r 2191, nf5ri 2193. (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 34171	. 2 ⊢ (Ⅎ'𝑥𝜑 ↔ ((∃𝑥𝜑 → 𝜑) ∧ (𝜑 → ∀𝑥𝜑)))
2	1	simprbi 500	1 ⊢ (Ⅎ'𝑥𝜑 → (𝜑 → ∀𝑥𝜑))

Syntax hints:   → wi 4  ∀wal 1536  ∃wex 1781  Ⅎ'wnnf 34170

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 210  df-an 400  df-bj-nnf 34171

This theorem is referenced by:  bj-nnfad  34176  bj-nnfea  34179  bj-nnfim1  34188  bj-nnfim2  34189  bj-nnf-alrim  34199  bj-19.23t  34214  bj-19.37im  34216  bj-19.42t  34217  bj-sbft  34219