Theorem bj-nnfe 37206

Description: Nonfreeness implies the equivalent of ax5e 1932. (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 37202	. 2 ⊢ (Ⅎ'𝑥𝜑 ↔ ((∃𝑥𝜑 → 𝜑) ∧ (𝜑 → ∀𝑥𝜑)))
2	1	simplbi 500	1 ⊢ (Ⅎ'𝑥𝜑 → (∃𝑥𝜑 → 𝜑))

Syntax hints:   → wi 4  ∀wal 1558  ∃wex 1799  Ⅎ'wnnf 37201

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 400  df-bj-nnf 37202

This theorem is referenced by:  bj-nnfed  37207  bj-nnfei  37208  bj-nnfea  37209  bj-nnfim1  37216  bj-nnfim2  37217  bj-nnf-exlim  37235  bj-19.21t  37236  bj-19.36im  37238  bj-19.42t  37240  bj-sbft  37253