Theorem nfnf 1571

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 1455	. 2 ⊢ (Ⅎ𝑦𝜑 ↔ ∀𝑦(𝜑 → ∀𝑦𝜑))
2		nfal.1	. . . 4 ⊢ Ⅎ𝑥𝜑
3	2	nfal 1570	. . . 4 ⊢ Ⅎ𝑥∀𝑦𝜑
4	2, 3	nfim 1566	. . 3 ⊢ Ⅎ𝑥(𝜑 → ∀𝑦𝜑)
5	4	nfal 1570	. 2 ⊢ Ⅎ𝑥∀𝑦(𝜑 → ∀𝑦𝜑)
6	1, 5	nfxfr 1468	1 ⊢ Ⅎ𝑥Ⅎ𝑦𝜑

Syntax hints:   → wi 4  ∀wal 1347  Ⅎwnf 1454

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1441  ax-7 1442  ax-gen 1443  ax-4 1504  ax-ial 1528  ax-i5r 1529

This theorem depends on definitions:  df-bi 116  df-nf 1455

This theorem is referenced by:  nfnfc  2320