Theorem bj-nnfa 34062

Description: Nonfreeness implies the equivalent of ax-5 1911. See nf5r 2193, nf5ri 2195. (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 34058	. 2 ⊢ (Ⅎ'𝑥𝜑 ↔ ((∃𝑥𝜑 → 𝜑) ∧ (𝜑 → ∀𝑥𝜑)))
2	1	simprbi 499	1 ⊢ (Ⅎ'𝑥𝜑 → (𝜑 → ∀𝑥𝜑))

Syntax hints:   → wi 4  ∀wal 1535  ∃wex 1780  Ⅎ'wnnf 34057

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 34058

This theorem is referenced by:  bj-nnfad  34063  bj-nnfea  34066  bj-nnfim1  34075  bj-nnfim2  34076  bj-nnf-alrim  34086  bj-19.23t  34101  bj-19.37im  34103  bj-19.42t  34104  bj-sbft  34106