Theorem nfsbv 2365

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 2557 with an additional disjoint variable condition on 𝑥, 𝑧 but not requiring ax-13 2406. (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 2234	. . 3 ⊢ (𝜑 → ∀𝑧𝜑)
3	2	hbsbw 2209	. 2 ⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑧[𝑦 / 𝑥]𝜑)
4	3	nf5i 2184	1 ⊢ Ⅎ𝑧[𝑦 / 𝑥]𝜑

This proof depends on syntax axioms:  Ⅎwnf 1816  [wsb 2099

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 1943  ax-6 2000  ax-7 2041  ax-10 2179  ax-11 2195  ax-12 2216

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-nf 1817  df-sb 2100

This theorem is used by:  sbco2v  2366  2sb8ef  2390  sb8euv  2629  2mo  2678  cbvrabcsfw  3895  cbvopab1  5187  cbvmptf  5213  ralxpf  5834  cbviotaw  6503  cbvriotaw  7385  dfoprab4f  8059  mo5f  32908  ax11-pm2  37530  dfich2  48267  ichbi12i  48269