Theorem nfs1 2493

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 2373. Check out nfs1v 2156 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 2191	. . 3 ⊢ (𝜑 → ∀𝑦𝜑)
3	2	hbsb3 2492	. 2 ⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑)
4	3	nf5i 2145	1 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑

Syntax hints:  Ⅎwnf 1789  [wsb 2070

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-10 2140  ax-12 2174  ax-13 2373

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-ex 1786  df-nf 1790  df-sb 2071

This theorem is referenced by:  sb8  2522  sb8e  2523