Theorem nfs1 1802

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 1512	. . 3 ⊢ (𝜑 → ∀𝑦𝜑)
3	2	hbsb3 1801	. 2 ⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑)
4	3	nfi 1455	1 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑

Syntax hints:  Ⅎwnf 1453  [wsb 1755

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 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-11 1499  ax-4 1503  ax-i9 1523  ax-ial 1527

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756

This theorem is referenced by:  sb8  1849  sb8e  1850