Theorem bj-nnfe 34086

Description: Nonfreeness implies the equivalent of ax5e 1912. (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 34080	. 2 ⊢ (Ⅎ'𝑥𝜑 ↔ ((∃𝑥𝜑 → 𝜑) ∧ (𝜑 → ∀𝑥𝜑)))
2	1	simplbi 500	1 ⊢ (Ⅎ'𝑥𝜑 → (∃𝑥𝜑 → 𝜑))

Syntax hints:   → wi 4  ∀wal 1534  ∃wex 1779  Ⅎ'wnnf 34079

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 34080

This theorem is referenced by:  bj-nnfed  34087  bj-nnfea  34088  bj-nnfim1  34097  bj-nnfim2  34098  bj-nnf-exlim  34109  bj-19.21t  34122  bj-19.36im  34124  bj-19.42t  34126  bj-sbft  34128