Theorem nfsbv 2323

Description: If 𝑧 is not free in 𝜑, then it is not free in [𝑦 / 𝑥]𝜑 when 𝑧 is disjoint from both 𝑥 and 𝑦. Version of nfsb 2522 with an additional disjoint variable condition on 𝑥, 𝑧 but not requiring ax-13 2371. (Contributed by Mario Carneiro, 11-Aug-2016.) (Revised by Wolf Lammen, 7-Feb-2023.) Remove disjoint variable condition on 𝑥, 𝑦. (Revised by Steven Nguyen, 13-Aug-2023.) (Proof shortened by Wolf Lammen, 25-Oct-2024.)

Hypothesis

Ref	Expression
nfsbv.nf	⊢ Ⅎ𝑧𝜑

Assertion

Ref	Expression
nfsbv	⊢ Ⅎ𝑧[𝑦 / 𝑥]𝜑

Distinct variable groups:   𝑥,𝑧   𝑦,𝑧

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

Proof of Theorem nfsbv

Step	Hyp	Ref	Expression
1		nfsbv.nf	. . . 4 ⊢ Ⅎ𝑧𝜑
2	1	nf5ri 2188	. . 3 ⊢ (𝜑 → ∀𝑧𝜑)
3	2	hbsbw 2169	. 2 ⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑧[𝑦 / 𝑥]𝜑)
4	3	nf5i 2142	1 ⊢ Ⅎ𝑧[𝑦 / 𝑥]𝜑

Syntax hints:  Ⅎwnf 1785  [wsb 2067

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-10 2137  ax-11 2154  ax-12 2171

This theorem depends on definitions:  df-bi 206  df-ex 1782  df-nf 1786  df-sb 2068

This theorem is referenced by:  hbsbwOLD  2325  sbco2v  2326  2sb8ef  2352  sb8euv  2593  2mo  2644  nfcriiOLD  2896  cbvralfwOLD  3320  cbvreuwOLD  3412  cbvrabw  3467  cbvrabcsfw  3936  cbvopab1  5222  cbvmptf  5256  ralxpf  5844  cbviotaw  6499  cbvriotaw  7370  dfoprab4f  8038  mo5f  31716  ax11-pm2  35702  dfich2  46112  ichbi12i  46114