Theorem nfs1f 1753

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 1499	. . 3 ⊢ (𝜑 → ∀𝑥𝜑)
3	2	sbh 1749	. 2 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜑)
4	3, 1	nfxfr 1450	1 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑

Syntax hints:  Ⅎwnf 1436  [wsb 1735

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 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-i9 1510  ax-ial 1514

This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736

This theorem is referenced by: (None)