Theorem nf5-1 2134

Description: One direction of nf5 2271 can be proved with a smaller footprint on axiom usage. (Contributed by Wolf Lammen, 16-Sep-2021.)

Assertion

Ref	Expression
nf5-1	⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → Ⅎ𝑥𝜑)

Proof of Theorem nf5-1

Step	Hyp	Ref	Expression
1		exim 1829	. . 3 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (∃𝑥𝜑 → ∃𝑥∀𝑥𝜑))
2		hbe1a 2133	. . 3 ⊢ (∃𝑥∀𝑥𝜑 → ∀𝑥𝜑)
3	1, 2	syl6 35	. 2 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (∃𝑥𝜑 → ∀𝑥𝜑))
4	3	nfd 1785	1 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → Ⅎ𝑥𝜑)

Syntax hints:   → wi 4  ∀wal 1532  ∃wex 1774  Ⅎwnf 1778

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-10 2130

This theorem depends on definitions:  df-bi 206  df-ex 1775  df-nf 1779

This theorem is referenced by:  nf5i  2135  nf5dh  2136  nf5d  2273  hbnt  2283  19.9ht  2308