Theorem nfs1f 1660

Description: If x is not free in φ, it is not free in [y / x]φ. (Contributed by Mario Carneiro, 11-Aug-2016.)

Hypothesis

Ref	Expression
nfs1f.1	⊢ Ⅎxφ

Assertion

Ref	Expression
nfs1f	⊢ Ⅎx[y / x]φ

Proof of Theorem nfs1f

Step	Hyp	Ref	Expression
1		nfs1f.1	. . . 4 ⊢ Ⅎxφ
2	1	nfri 1409	. . 3 ⊢ (φ → ∀xφ)
3	2	sbh 1656	. 2 ⊢ ([y / x]φ ↔ φ)
4	3, 1	nfxfr 1360	1 ⊢ Ⅎx[y / x]φ

Syntax hints:  Ⅎwnf 1346  [wsb 1642

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-4 1397  ax-i9 1420  ax-ial 1424

This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643

This theorem is referenced by: (None)