Theorem bj-nfdt0 34877

Description: A theorem close to a closed form of nf5d 2281 and nf5dh 2143. (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 1813	. 2 ⊢ (∀𝑥(𝜑 → (𝜓 → ∀𝑥𝜓)) → (∀𝑥𝜑 → ∀𝑥(𝜓 → ∀𝑥𝜓)))
2		nf5 2279	. 2 ⊢ (Ⅎ𝑥𝜓 ↔ ∀𝑥(𝜓 → ∀𝑥𝜓))
3	1, 2	syl6ibr 251	1 ⊢ (∀𝑥(𝜑 → (𝜓 → ∀𝑥𝜓)) → (∀𝑥𝜑 → Ⅎ𝑥𝜓))

Syntax hints:   → wi 4  ∀wal 1537  Ⅎwnf 1786

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-10 2137  ax-12 2171

This theorem depends on definitions:  df-bi 206  df-or 845  df-ex 1783  df-nf 1787

This theorem is referenced by:  bj-nfdt  34878