Theorem bj-nnfe 37279

Description: Nonfreeness implies the equivalent of ax5e 1939. (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 37275	. 2 ⊢ (Ⅎ'𝑥𝜑 ↔ ((∃𝑥𝜑 → 𝜑) ∧ (𝜑 → ∀𝑥𝜑)))
2	1	simplbi 501	1 ⊢ (Ⅎ'𝑥𝜑 → (∃𝑥𝜑 → 𝜑))

Syntax hints:   → wi 4  ∀wal 1565  ∃wex 1806  Ⅎ'wnnf 37274

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 401  df-bj-nnf 37275

This theorem is referenced by:  bj-nnfed  37280  bj-nnfei  37281  bj-nnfea  37282  bj-nnfim1  37289  bj-nnfim2  37290  bj-nnf-exlim  37308  bj-19.21t  37309  bj-19.36im  37311  bj-19.42t  37313  bj-sbft  37326