Theorem nfs1f 2312

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	. . 3 ⊢ Ⅎ𝑥𝜑
2	1	sbf 2308	. 2 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜑)
3	2, 1	nfxfr 1886	1 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑

This proof depends on syntax axioms:  Ⅎwnf 1816  [wsb 2101

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-12 2216

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-nf 1817  df-sb 2102

This theorem is used by: (None)