Theorem bj-nnfe 37417

Description: Nonfreeness implies the equivalent of ax5e 1945. (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 37413	. 2 ⊢ (Ⅎ'𝑥𝜑 ↔ ((∃𝑥𝜑 → 𝜑) ∧ (𝜑 → ∀𝑥𝜑)))
2	1	simplbi 502	1 ⊢ (Ⅎ'𝑥𝜑 → (∃𝑥𝜑 → 𝜑))

This proof depends on syntax axioms:   → wi 4  ∀wal 1568  ∃wex 1812  Ⅎ'wnnf 37412

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-an 402  df-bj-nnf 37413

This theorem is used by:  bj-nnfed  37418  bj-nnfei  37419  bj-nnfea  37420  bj-nnfim1  37427  bj-nnfim2  37428  bj-nnf-exlim  37446  bj-19.21t  37447  bj-19.36im  37449  bj-19.42t  37451  bj-sbft  37464