Theorem bj-nnfad 37413

Description: Nonfreeness implies the equivalent of ax-5 1943, deduction form. See nf5rd 2235. (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 37412	. 2 ⊢ (Ⅎ'𝑥𝜓 → (𝜓 → ∀𝑥𝜓))
3	1, 2	syl 18	1 ⊢ (𝜑 → (𝜓 → ∀𝑥𝜓))

This proof depends on syntax axioms:   → wi 4  ∀wal 1568  Ⅎ'wnnf 37410

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 37411

This theorem is used by:  bj-nnfand  37439  bj-nnford  37441  bj-nnf-cbvali  37463