Theorem nfald 1852

Description: If x is not free in φ, it is not free in ∀yφ. (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 6-Jan-2018.)

Hypotheses

Ref	Expression
nfald.1	⊢ Ⅎyφ
nfald.2	⊢ (φ → Ⅎxψ)

Assertion

Ref	Expression
nfald	⊢ (φ → Ⅎx∀yψ)

Proof of Theorem nfald

Step	Hyp	Ref	Expression
1		nfald.1	. . 3 ⊢ Ⅎyφ
2		nfald.2	. . 3 ⊢ (φ → Ⅎxψ)
3	1, 2	alrimi 1765	. 2 ⊢ (φ → ∀yℲxψ)
4		nfnf1 1790	. . . 4 ⊢ ℲxℲxψ
5	4	nfal 1842	. . 3 ⊢ Ⅎx∀yℲxψ
6		hba1 1786	. . . 4 ⊢ (∀yℲxψ → ∀y∀yℲxψ)
7		sp 1747	. . . . 5 ⊢ (∀yℲxψ → Ⅎxψ)
8	7	nfrd 1763	. . . 4 ⊢ (∀yℲxψ → (ψ → ∀xψ))
9	6, 8	hbald 1740	. . 3 ⊢ (∀yℲxψ → (∀yψ → ∀x∀yψ))
10	5, 9	nfd 1766	. 2 ⊢ (∀yℲxψ → Ⅎx∀yψ)
11	3, 10	syl 15	1 ⊢ (φ → Ⅎx∀yψ)

Syntax hints:   → wi 4  ∀wal 1540  Ⅎwnf 1544

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-7 1734  ax-11 1746

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

This theorem is referenced by:  nfexd  1854  nfald2  1972  nfsb4t  2080  nfeqd  2504