Theorem bj-nnfai 36731

Description: Nonfreeness implies the equivalent of ax-5 1910, inference form. See nf5ri 2195. (Contributed by BJ, 22-Sep-2024.)

Hypothesis

Ref	Expression
bj-nnfai.1	⊢ Ⅎ'𝑥𝜑

Assertion

Ref	Expression
bj-nnfai	⊢ (𝜑 → ∀𝑥𝜑)

Proof of Theorem bj-nnfai

Step	Hyp	Ref	Expression
1		bj-nnfai.1	. 2 ⊢ Ⅎ'𝑥𝜑
2		bj-nnfa 36729	. 2 ⊢ (Ⅎ'𝑥𝜑 → (𝜑 → ∀𝑥𝜑))
3	1, 2	ax-mp 5	1 ⊢ (𝜑 → ∀𝑥𝜑)

Syntax hints:   → wi 4  ∀wal 1538  Ⅎ'wnnf 36724

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 207  df-an 396  df-bj-nnf 36725

This theorem is referenced by: (None)