Theorem nfsbv 2362

Description: If 𝑧 is not free in 𝜑, it is not free in [𝑦 / 𝑥]𝜑 when 𝑥, 𝑦 and 𝑧 are distinct. Version of nfsb 2575 requiring more disjoint variables, but fewer axioms. (Contributed by Wolf Lammen, 7-Feb-2023.)

Hypothesis

Ref	Expression
nfsbv.nf	⊢ Ⅎ𝑧𝜑

Assertion

Ref	Expression
nfsbv	⊢ Ⅎ𝑧[𝑦 / 𝑥]𝜑

Distinct variable group:   𝑥,𝑦,𝑧

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)

Proof of Theorem nfsbv

Step	Hyp	Ref	Expression
1		sb6 2307	. 2 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑))
2		nfv 2013	. . . 4 ⊢ Ⅎ𝑧 𝑥 = 𝑦
3		nfsbv.nf	. . . 4 ⊢ Ⅎ𝑧𝜑
4	2, 3	nfim 1999	. . 3 ⊢ Ⅎ𝑧(𝑥 = 𝑦 → 𝜑)
5	4	nfal 2355	. 2 ⊢ Ⅎ𝑧∀𝑥(𝑥 = 𝑦 → 𝜑)
6	1, 5	nfxfr 1952	1 ⊢ Ⅎ𝑧[𝑦 / 𝑥]𝜑

Syntax hints:   → wi 4  ∀wal 1654  Ⅎwnf 1882  [wsb 2067

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-10 2192  ax-11 2207  ax-12 2220

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-ex 1879  df-nf 1883  df-sb 2068

This theorem is referenced by:  2sb8ev  2378  sb8euv  2687