Theorem nfs1 2044

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

Hypothesis

Ref	Expression
nfs1.1	⊢ Ⅎyφ

Assertion

Ref	Expression
nfs1	⊢ Ⅎx[y / x]φ

Proof of Theorem nfs1

Step	Hyp	Ref	Expression
1		nfs1.1	. . . 4 ⊢ Ⅎyφ
2	1	nfri 1762	. . 3 ⊢ (φ → ∀yφ)
3	2	hbsb3 2043	. 2 ⊢ ([y / x]φ → ∀x[y / x]φ)
4	3	nfi 1551	1 ⊢ Ⅎx[y / x]φ

Syntax hints:  Ⅎwnf 1544  [wsb 1648

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  ax-12 1925

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649

This theorem is referenced by:  ax16ALT2  2048  sbco2  2086  sb8  2092  mo  2226  ralxpf  4827