Theorem bj-nnfea 34843

Description: Nonfreeness implies the equivalent of ax5ea 1917. (Contributed by BJ, 28-Jul-2023.)

Assertion

Ref	Expression
bj-nnfea	⊢ (Ⅎ'𝑥𝜑 → (∃𝑥𝜑 → ∀𝑥𝜑))

Proof of Theorem bj-nnfea

Step	Hyp	Ref	Expression
1		bj-nnfe 34840	. 2 ⊢ (Ⅎ'𝑥𝜑 → (∃𝑥𝜑 → 𝜑))
2		bj-nnfa 34837	. 2 ⊢ (Ⅎ'𝑥𝜑 → (𝜑 → ∀𝑥𝜑))
3	1, 2	syld 47	1 ⊢ (Ⅎ'𝑥𝜑 → (∃𝑥𝜑 → ∀𝑥𝜑))

Syntax hints:   → wi 4  ∀wal 1537  ∃wex 1783  Ⅎ'wnnf 34832

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 206  df-an 396  df-bj-nnf 34833

This theorem is referenced by:  bj-nnfead  34844  bj-nnfeai  34845  bj-nnfnfTEMP  34847  bj-dfnnf3  34866