Theorem hbimd 1815

Description: Deduction form of bound-variable hypothesis builder hbim 1817. (Contributed by NM, 1-Jan-2002.) (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	nfdh 1767	. . 3 ⊢ (φ → Ⅎxψ)
4		hbimd.3	. . . 4 ⊢ (φ → (χ → ∀xχ))
5	1, 4	nfdh 1767	. . 3 ⊢ (φ → Ⅎxχ)
6	3, 5	nfimd 1808	. 2 ⊢ (φ → Ⅎx(ψ → χ))
7	6	nfrd 1763	1 ⊢ (φ → ((ψ → χ) → ∀x(ψ → χ)))

Syntax hints:   → wi 4  ∀wal 1540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-11 1746

This theorem depends on definitions:  df-bi 177  df-ex 1542  df-nf 1545

This theorem is referenced by:  spimehOLD  1821  dvelimhw  1849  dvelimv  1939  dvelimh  1964  dvelimALT  2133  dvelimf-o  2180