Theorem nf5-1 2051

Description: One direction of nf5 . (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 1621	. . 3 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (∃𝑥𝜑 → ∃𝑥∀𝑥𝜑))
2		hbe1a 2050	. . 3 ⊢ (∃𝑥∀𝑥𝜑 → ∀𝑥𝜑)
3	1, 2	syl6 33	. 2 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (∃𝑥𝜑 → ∀𝑥𝜑))
4	3	nfd2 2049	1 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → Ⅎ𝑥𝜑)

Syntax hints:   → wi 4  ∀wal 1370  Ⅎwnf 1482  ∃wex 1514

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1469  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-4 1532  ax-ial 1556  ax-i5r 1557

This theorem depends on definitions:  df-bi 117  df-nf 1483

This theorem is referenced by:  nf5d  2052