Theorem bj-nnfad 34911

Description: Nonfreeness implies the equivalent of ax-5 1913, deduction form. See nf5rd 2189. (Contributed by BJ, 2-Dec-2023.)

Hypothesis

Ref	Expression
bj-nnfad.1	⊢ (𝜑 → Ⅎ'𝑥𝜓)

Assertion

Ref	Expression
bj-nnfad	⊢ (𝜑 → (𝜓 → ∀𝑥𝜓))

Proof of Theorem bj-nnfad

Step	Hyp	Ref	Expression
1		bj-nnfad.1	. 2 ⊢ (𝜑 → Ⅎ'𝑥𝜓)
2		bj-nnfa 34910	. 2 ⊢ (Ⅎ'𝑥𝜓 → (𝜓 → ∀𝑥𝜓))
3	1, 2	syl 17	1 ⊢ (𝜑 → (𝜓 → ∀𝑥𝜓))

Syntax hints:   → wi 4  ∀wal 1537  Ⅎ'wnnf 34905

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 397  df-bj-nnf 34906

This theorem is referenced by:  bj-nnfand  34931  bj-nnford  34933