Theorem hbimd 2272

Description: Deduction form of bound-variable hypothesis builder hbim 2273. (Contributed by NM, 14-May-1993.) (Proof shortened by Wolf Lammen, 3-Jan-2018.)

Hypotheses

Ref	Expression
hbimd.1	⊢ (𝜑 → ∀𝑥𝜑)
hbimd.2	⊢ (𝜑 → (𝜓 → ∀𝑥𝜓))
hbimd.3	⊢ (𝜑 → (𝜒 → ∀𝑥𝜒))

Assertion

Ref	Expression
hbimd	⊢ (𝜑 → ((𝜓 → 𝜒) → ∀𝑥(𝜓 → 𝜒)))

Proof of Theorem hbimd

Step	Hyp	Ref	Expression
1		hbimd.1	. . . 4 ⊢ (𝜑 → ∀𝑥𝜑)
2		hbimd.2	. . . 4 ⊢ (𝜑 → (𝜓 → ∀𝑥𝜓))
3	1, 2	nf5dh 2140	. . 3 ⊢ (𝜑 → Ⅎ𝑥𝜓)
4		hbimd.3	. . . 4 ⊢ (𝜑 → (𝜒 → ∀𝑥𝜒))
5	1, 4	nf5dh 2140	. . 3 ⊢ (𝜑 → Ⅎ𝑥𝜒)
6	3, 5	nfimd 1940	. 2 ⊢ (𝜑 → Ⅎ𝑥(𝜓 → 𝜒))
7	6	nf5rd 2180	1 ⊢ (𝜑 → ((𝜓 → 𝜒) → ∀𝑥(𝜓 → 𝜒)))

Syntax hints:   → wi 4  ∀wal 1599

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-10 2134  ax-12 2162

This theorem depends on definitions:  df-bi 199  df-ex 1824  df-nf 1828

This theorem is referenced by:  dvelimf-o  35077