Theorem nf5di 2322

Description: Since the converse holds by a1i 11, this inference shows that we can represent a not-free hypothesis with either Ⅎ𝑥𝜑 (inference form) or (𝜑 → Ⅎ𝑥𝜑) (deduction form). (Contributed by NM, 17-Aug-2018.) (Proof shortened by Wolf Lammen, 10-Jul-2019.)

Hypothesis

Ref	Expression
nf5di.1	⊢ (𝜑 → Ⅎ𝑥𝜑)

Assertion

Ref	Expression
nf5di	⊢ Ⅎ𝑥𝜑

Proof of Theorem nf5di

Step	Hyp	Ref	Expression
1		nf5di.1	. . . 4 ⊢ (𝜑 → Ⅎ𝑥𝜑)
2	1	nf5rd 2235	. . 3 ⊢ (𝜑 → (𝜑 → ∀𝑥𝜑))
3	2	pm2.43i 53	. 2 ⊢ (𝜑 → ∀𝑥𝜑)
4	3	nf5i 2184	1 ⊢ Ⅎ𝑥𝜑

This proof depends on syntax axioms:   → wi 4  ∀wal 1568  Ⅎwnf 1816

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-10 2179  ax-12 2216

This proof depends on definitions:  df-bi 210  df-ex 1813  df-nf 1817

This theorem is used by: (None)