Theorem bj-nnfea 37766

Description: Nonfreeness implies the equivalent of ax5ea 1947. (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 37763	. 2 ⊢ (Ⅎ'𝑥𝜑 → (∃𝑥𝜑 → 𝜑))
2		bj-nnfa 37760	. 2 ⊢ (Ⅎ'𝑥𝜑 → (𝜑 → ∀𝑥𝜑))
3	1, 2	syld 48	1 ⊢ (Ⅎ'𝑥𝜑 → (∃𝑥𝜑 → ∀𝑥𝜑))

This proof depends on syntax axioms:   → wi 4  ∀wal 1567  ∃wex 1811  Ⅎ'wnnf 37758

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 37759

This theorem is used by:  bj-nnfead  37767  bj-nnfeai  37768  bj-nnfnfTEMP  37772  bj-dfnnf3  37813