Theorem nf5-1 2181

Description: One direction of nf5 2315 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 1866	. . 3 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (∃𝑥𝜑 → ∃𝑥∀𝑥𝜑))
2		hbe1a 2180	. . 3 ⊢ (∃𝑥∀𝑥𝜑 → ∀𝑥𝜑)
3	1, 2	syl6 36	. 2 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (∃𝑥𝜑 → ∀𝑥𝜑))
4	3	nfd 1822	1 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → Ⅎ𝑥𝜑)

This proof depends on syntax axioms:   → wi 4  ∀wal 1567  ∃wex 1811  Ⅎwnf 1815

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-10 2177

This proof depends on definitions:  df-bi 210  df-ex 1812  df-nf 1816

This theorem is used by:  nf5i  2182  nf5dh  2183  nf5d  2317  hbnt  2327  19.9ht  2350