Theorem bj-nfdt0 37689

Description: A theorem close to a closed form of nf5d 2317 and nf5dh 2183. (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 1842	. 2 ⊢ (∀𝑥(𝜑 → (𝜓 → ∀𝑥𝜓)) → (∀𝑥𝜑 → ∀𝑥(𝜓 → ∀𝑥𝜓)))
2		nf5 2315	. 2 ⊢ (Ⅎ𝑥𝜓 ↔ ∀𝑥(𝜓 → ∀𝑥𝜓))
3	1, 2	imbitrrdi 255	1 ⊢ (∀𝑥(𝜑 → (𝜓 → ∀𝑥𝜓)) → (∀𝑥𝜑 → Ⅎ𝑥𝜓))

This proof depends on syntax axioms:   → wi 4  ∀wal 1567  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-5 1942  ax-6 1999  ax-7 2040  ax-10 2177  ax-12 2212

This proof depends on definitions:  df-bi 210  df-or 862  df-ex 1812  df-nf 1816

This theorem is used by:  bj-nfdt  37690