Theorem bj-nfdt0 37182

Description: A theorem close to a closed form of nf5d 2321 and nf5dh 2184. (Contributed by BJ, 2-May-2019.)

Assertion

Ref	Expression
bj-nfdt0	⊢ (∀𝑥(𝜑 → (𝜓 → ∀𝑥𝜓)) → (∀𝑥𝜑 → Ⅎ𝑥𝜓))

Proof of Theorem bj-nfdt0

Step	Hyp	Ref	Expression
1		alim 1833	. 2 ⊢ (∀𝑥(𝜑 → (𝜓 → ∀𝑥𝜓)) → (∀𝑥𝜑 → ∀𝑥(𝜓 → ∀𝑥𝜓)))
2		nf5 2319	. 2 ⊢ (Ⅎ𝑥𝜓 ↔ ∀𝑥(𝜓 → ∀𝑥𝜓))
3	1, 2	imbitrrdi 255	1 ⊢ (∀𝑥(𝜑 → (𝜓 → ∀𝑥𝜓)) → (∀𝑥𝜑 → Ⅎ𝑥𝜓))

Syntax hints:   → wi 4  ∀wal 1561  Ⅎwnf 1806

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-10 2178  ax-12 2215

This theorem depends on definitions:  df-bi 210  df-or 861  df-ex 1803  df-nf 1807

This theorem is referenced by:  bj-nfdt  37183