Theorem nfnf 1845

Description: If x is not free in φ, it is not free in Ⅎyφ. (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 30-Dec-2017.)

Hypothesis

Ref	Expression
nfal.1	⊢ Ⅎxφ

Assertion

Ref	Expression
nfnf	⊢ ℲxℲyφ

Proof of Theorem nfnf

Step	Hyp	Ref	Expression
1		df-nf 1545	. 2 ⊢ (Ⅎyφ ↔ ∀y(φ → ∀yφ))
2		nfal.1	. . . 4 ⊢ Ⅎxφ
3	2	nfal 1842	. . . 4 ⊢ Ⅎx∀yφ
4	2, 3	nfim 1813	. . 3 ⊢ Ⅎx(φ → ∀yφ)
5	4	nfal 1842	. 2 ⊢ Ⅎx∀y(φ → ∀yφ)
6	1, 5	nfxfr 1570	1 ⊢ ℲxℲyφ

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:  nfnfc  2495