Theorem bj-nnfeai 37389

Description: Nonfreeness implies the equivalent of ax5ea 1942, inference form. (Contributed by BJ, 22-Sep-2024.)

Hypothesis

Ref	Expression
bj-nnfeai.1	⊢ Ⅎ'𝑥𝜑

Assertion

Ref	Expression
bj-nnfeai	⊢ (∃𝑥𝜑 → ∀𝑥𝜑)

Proof of Theorem bj-nnfeai

Step	Hyp	Ref	Expression
1		bj-nnfeai.1	. 2 ⊢ Ⅎ'𝑥𝜑
2		bj-nnfea 37387	. 2 ⊢ (Ⅎ'𝑥𝜑 → (∃𝑥𝜑 → ∀𝑥𝜑))
3	1, 2	ax-mp 5	1 ⊢ (∃𝑥𝜑 → ∀𝑥𝜑)

This proof depends on syntax axioms:   → wi 4  ∀wal 1567  ∃wex 1808  Ⅎ'wnnf 37379

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 401  df-bj-nnf 37380

This theorem is used by: (None)