Theorem bj-nnfe 36875

Description: Nonfreeness implies the equivalent of ax5e 1913. (Contributed by BJ, 28-Jul-2023.)

Assertion

Ref	Expression
bj-nnfe	⊢ (Ⅎ'𝑥𝜑 → (∃𝑥𝜑 → 𝜑))

Proof of Theorem bj-nnfe

Step	Hyp	Ref	Expression
1		df-bj-nnf 36868	. 2 ⊢ (Ⅎ'𝑥𝜑 ↔ ((∃𝑥𝜑 → 𝜑) ∧ (𝜑 → ∀𝑥𝜑)))
2	1	simplbi 497	1 ⊢ (Ⅎ'𝑥𝜑 → (∃𝑥𝜑 → 𝜑))

Syntax hints:   → wi 4  ∀wal 1539  ∃wex 1780  Ⅎ'wnnf 36867

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 207  df-an 396  df-bj-nnf 36868

This theorem is referenced by:  bj-nnfed  36876  bj-nnfei  36877  bj-nnfea  36878  bj-nnfim1  36888  bj-nnfim2  36889  bj-nnf-exlim  36900  bj-19.21t  36913  bj-19.36im  36915  bj-19.42t  36917  bj-sbft  36919