Theorem nf5di 2184

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.) Reduce axiom dependencies. (Revised by SN and Sky Winter, 23-Oct-2026.)

Hypothesis

Ref	Expression
nf5di.1	⊢ (𝜑 → Ⅎ𝑥𝜑)

Assertion

Ref	Expression
nf5di	⊢ Ⅎ𝑥𝜑

Proof of Theorem nf5di

Step	Hyp	Ref	Expression
1		hbe1 2182	. . . . 5 ⊢ (∃𝑥𝜑 → ∀𝑥∃𝑥𝜑)
2		nf5di.1	. . . . . . 7 ⊢ (𝜑 → Ⅎ𝑥𝜑)
3	2	nfrd 1824	. . . . . 6 ⊢ (𝜑 → (∃𝑥𝜑 → ∀𝑥𝜑))
4	3	com12 33	. . . . 5 ⊢ (∃𝑥𝜑 → (𝜑 → ∀𝑥𝜑))
5	1, 4	eximdh 1897	. . . 4 ⊢ (∃𝑥𝜑 → (∃𝑥𝜑 → ∃𝑥∀𝑥𝜑))
6	5	pm2.43i 53	. . 3 ⊢ (∃𝑥𝜑 → ∃𝑥∀𝑥𝜑)
7		hbe1a 2183	. . 3 ⊢ (∃𝑥∀𝑥𝜑 → ∀𝑥𝜑)
8	6, 7	syl 18	. 2 ⊢ (∃𝑥𝜑 → ∀𝑥𝜑)
9	8	nfi 1821	1 ⊢ Ⅎ𝑥𝜑

This proof depends on syntax axioms:   → wi 4  ∀wal 1568  ∃wex 1812  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-10 2180

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

This theorem is used by: (None)