Theorem bj-nnfe 34551

Description: Nonfreeness implies the equivalent of ax5e 1919. (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 34544	. 2 ⊢ (Ⅎ'𝑥𝜑 ↔ ((∃𝑥𝜑 → 𝜑) ∧ (𝜑 → ∀𝑥𝜑)))
2	1	simplbi 501	1 ⊢ (Ⅎ'𝑥𝜑 → (∃𝑥𝜑 → 𝜑))

Syntax hints:   → wi 4  ∀wal 1540  ∃wex 1786  Ⅎ'wnnf 34543

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 34544

This theorem is referenced by:  bj-nnfed  34552  bj-nnfei  34553  bj-nnfea  34554  bj-nnfim1  34564  bj-nnfim2  34565  bj-nnf-exlim  34576  bj-19.21t  34589  bj-19.36im  34591  bj-19.42t  34593  bj-sbft  34595