Theorem nfsbv 2331

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 2523 with an additional disjoint variable condition on 𝑥, 𝑧 but not requiring ax-13 2372. (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 2198	. . 3 ⊢ (𝜑 → ∀𝑧𝜑)
3	2	hbsbw 2174	. 2 ⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑧[𝑦 / 𝑥]𝜑)
4	3	nf5i 2149	1 ⊢ Ⅎ𝑧[𝑦 / 𝑥]𝜑

Syntax hints:  Ⅎwnf 1784  [wsb 2067

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 1911  ax-6 1968  ax-7 2009  ax-10 2144  ax-11 2160  ax-12 2180

This theorem depends on definitions:  df-bi 207  df-ex 1781  df-nf 1785  df-sb 2068

This theorem is referenced by:  sbco2v  2332  2sb8ef  2356  sb8euv  2594  2mo  2643  cbvrabwOLD  3431  cbvrabcsfw  3886  cbvopab1  5163  cbvmptf  5189  ralxpf  5785  cbviotaw  6444  cbvriotaw  7312  dfoprab4f  7988  mo5f  32468  ax11-pm2  36880  dfich2  47568  ichbi12i  47570