Theorem nfnf 1509

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 1390	. 2 ⊢ (Ⅎ𝑦𝜑 ↔ ∀𝑦(𝜑 → ∀𝑦𝜑))
2		nfal.1	. . . 4 ⊢ Ⅎ𝑥𝜑
3	2	nfal 1508	. . . 4 ⊢ Ⅎ𝑥∀𝑦𝜑
4	2, 3	nfim 1504	. . 3 ⊢ Ⅎ𝑥(𝜑 → ∀𝑦𝜑)
5	4	nfal 1508	. 2 ⊢ Ⅎ𝑥∀𝑦(𝜑 → ∀𝑦𝜑)
6	1, 5	nfxfr 1403	1 ⊢ Ⅎ𝑥Ⅎ𝑦𝜑

Syntax hints:   → wi 4  ∀wal 1282  Ⅎwnf 1389

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-7 1377  ax-gen 1378  ax-4 1440  ax-ial 1467  ax-i5r 1468

This theorem depends on definitions:  df-bi 115  df-nf 1390

This theorem is referenced by:  nfnfc  2225