Theorem nfs1 2491

Description: If 𝑦 is not free in 𝜑, 𝑥 is not free in [𝑦 / 𝑥]𝜑. Usage of this theorem is discouraged because it depends on ax-13 2371. Check out nfs1v 2152 for a version requiring fewer axioms. (Contributed by Mario Carneiro, 11-Aug-2016.) (New usage is discouraged.)

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	nf5ri 2187	. . 3 ⊢ (𝜑 → ∀𝑦𝜑)
3	2	hbsb3 2490	. 2 ⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑)
4	3	nf5i 2141	1 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑

Syntax hints:  Ⅎwnf 1784  [wsb 2066

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-10 2136  ax-12 2170  ax-13 2371

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-ex 1781  df-nf 1785  df-sb 2067

This theorem is referenced by:  sb8  2520  sb8e  2521